Dynamics and Control of Electromagnetic Satellite Formations

Transcription

Dynamics and Control of Electromagnetic
Satellite Formations
Umair Ahsun, David W. Miller
June 2007
SSL # 12-07
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Dynamics and Control of Electromagnetic
Satellite Formations
by
Umair Ahsun
B.Sc. in Electrical Engineering
University of Engineering & Technology, Lahore Pakistan (1996)
S.M. in Aeronautical and Astronautical Engineering
Massachusetts Institute of Technology, Feb. 2004
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
AERONAUTICAL AND ASTRONAUTICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2007
© Massachusetts Institute of Technology 2007. All rights reserved.
Author________________________________________________________________________________
Graduate Research Assistant
Space Systems Laboratory
Certified By ___________________________________________________________________________
David W. Miller
Professor, Aeronautics & Astronautics
Thesis Committee Chair/Thesis Advisor
Certified By ___________________________________________________________________________
Jonathan P. How
Associate Professor, Aeronautics & Astronautics
Thesis Committee Co-Chair
Certified By ___________________________________________________________________________
James D. Paduano
Research Affiliate, Aeronautics & Astronautics Department
Thesis Committee Member
Certified By ___________________________________________________________________________
Raymond J. Sedwick
Principle Research Scientist, Aeronautics & Astronautics
Thesis Committee Member
Accepted By __________________________________________________________________________
Jaime Peraire
Professor, Aeronautics & Astronautics
Chair, Committee on Graduate Students
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Dynamics and Control of Electromagnetic Satellite Formations
by
Umair Ahsun
Abstract
Satellite formation flying is an enabling technology for many space missions, especially
for space-based telescopes. Usually there is a tight formation-keeping requirement that
may need constant expenditure of fuel or at least fuel is required for formation
reconfiguration. Electromagnetic Formation Flying (EMFF) is a novel concept that uses
superconducting electromagnetic coils to provide forces and torques between different
satellites in a formation which enables the control of all the relative degrees of freedom.
With EMFF, the life-span of the mission becomes independent of the fuel available on
board. Also the contamination of optics or sensitive formation instruments, due to
thruster plumes, is avoided. This comes at the cost of coupled and nonlinear dynamics of
the formation and makes the control problem a challenging one. In this thesis, the
dynamics for a general N-satellite electromagnetic formation will be derived for both
deep space missions and Low Earth Orbit (LEO) formations. Nonlinear control laws
using adaptive techniques will be derived for general formations in LEO. Angular
momentum management in LEO is a problem for EMFF due to interaction of the
magnetic dipoles with the Earth’s magnetic field. A solution of this problem for general
Electromagnetic (EM) formations will be presented in the form of a dipole polarity
switching control law. For EMFF, the formation reconfiguration problem is a nonlinear
and constrained optimal time control problem as fuel cost for EMFF is zero. Two
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different methods of trajectory generation, namely feedback motion planning using the
Artificial Potential Function Method (APFM) and optimal trajectory generation using the
Legendre Pseudospectral method, will be derived for general EM Formations. The results
of these methods are compared for random EM Formations. This comparison shows that
the artificial potential function method is a promising technique for solving the real-time
motion planning problem of nonlinear and constrained systems, such as EMFF, with low
computational cost. Specifically it is the purpose of this thesis to show that a fullyactuated N-satellite EM formation can be stabilized and controlled under fairly general
assumptions, therefore showing the viability of this novel approach for satellite formation
flying from a dynamics and controls perspective.
Thesis Supervisor: David W. Miller
Title: Professor Aeronautics & Astronautics Department
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Acknowledgements
I would like to acknowledge my advisor Dave Miller for his constant support and
guidance during my stay at Space Systems Lab. I would also like to thank my committee
members Jonathan How, Jim Paduano and Raymond Sedwick for their helpful
suggestions and critique that helped me to improve my research. I would like to thank my
mother as no doubt her love and prayers helped me remain focused during my studies.
And lastly and most importantly I thank my loving wife Sarah and our beloved son
Arslan for their support and understanding.
This work was partially funded under Contract FA9453-05-C-0180 from the Defense
Advanced Research Project Agency with Mr. Scott Franke from the US Air Force
Research Laboratory at Kirtland AFB, NM serving as the Project Officer. Partial funding
was also provided by a subcontract with Payload Systems Inc. in support of NASA SBIR
Contract NNG06CA12C, managed by Goddard Space Flight Center with Dr. Jesse
Leitner as the COTR.
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Table of Contents
ABSTRACT .................................................................................................................................................. 3
ACKNOWLEDGEMENTS ......................................................................................................................... 5
TABLE OF CONTENTS ............................................................................................................................. 7
LIST OF FIGURES.................................................................................................................................... 11
CHAPTER 1
INTRODUCTION ......................................................................................................... 13
1.1. MOTIVATION AND BACKGROUND ...................................................................................................... 13
1.2. OVERVIEW OF EMFF AND REVIEW OF PREVIOUS WORK ON EMFF .................................................. 18
1.3. RESEARCH OBJECTIVES AND APPROACH ........................................................................................... 20
1.4. THESIS OVERVIEW ............................................................................................................................. 22
CHAPTER 2
ELECTROMAGNETIC FORMATION DYNAMICS............................................... 25
2.1. TRANSLATIONAL DYNAMIC MODEL FOR LEO .................................................................................. 25
2.1.1. Preliminaries............................................................................................................................. 26
2.1.2. Derivation of Translational Equations of Motion ..................................................................... 28
2.2. ATTITUDE DYNAMICS INCLUDING GYRO-STIFFENING EFFECTS ........................................................ 37
2.2.1. Magnetic Torque ....................................................................................................................... 40
2.3. EARTH’S MAGNETIC FIELD MODEL ................................................................................................... 42
2.3.1. Tilted Dipole Model of the Earth’s Magnetic Field Main Component ..................................... 43
2.3.2. Full Representation of the Main Field using Spherical Harmonic Analysis............................. 46
2.3.3. Disturbance Force due to Earth’s Magnetic Field ................................................................... 48
2.4. DEEP SPACE 2D DYNAMICS ............................................................................................................... 49
2.5. RELATIVE TRANSLATIONAL DYNAMICS ............................................................................................ 54
2.6. SUMMARY ......................................................................................................................................... 58
CHAPTER 3
CONTROL OF EM FORMATIONS IN LEO ............................................................ 59
3.1. CONTROL FRAMEWORK ..................................................................................................................... 60
3.2. TRANSLATIONAL CONTROL FOR N-SATELLITE EM FORMATION ....................................................... 62
3.2.1. Proof of asymptotic convergence using Barbalat’s Lemma...................................................... 67
3.3. ADAPTIVE ATTITUDE CONTROL ........................................................................................................ 69
3.4. ANGULAR MOMENTUM MANAGEMENT ............................................................................................. 71
3.4.1. AMM for two-Satellite Formation using Sinusoidal Excitation ................................................ 72
3.4.2. AMM using Dipole Polarity Switching ..................................................................................... 73
3.5. SIMULATION RESULTS ....................................................................................................................... 75
3.5.1. Two-Satellite Cross Track Formation....................................................................................... 75
3.6. SUMMARY ......................................................................................................................................... 79
CHAPTER 4
APFM AND STABILITY ANALYSIS......................................................................... 81
4.1. ARTIFICIAL POTENTIAL FUNCTION METHOD FOR VEHICLE GUIDANCE AND CONTROL ..................... 81
4.1.1. An Illustrative Example............................................................................................................. 82
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4.1.2. Feedback Motion Planning using APFM and its Limitations ................................................... 87
4.2. STABILITY ANALYSIS OF EM FORMATIONS ....................................................................................... 91
4.2.1. Limitations of Linear Analysis of EMFF................................................................................... 91
4.3. RECONFIGURATION OF EM FORMATIONS USING APFM .................................................................... 94
4.3.1. Simulation Results................................................................................................................... 102
4.3.2. Computational Complexity of the Algorithm........................................................................... 106
4.4. POTENTIAL FUNCTION SHAPING FOR REDUCING MANEUVER TIME ................................................. 107
4.4.1. Two-Satellite Formation Simulation Results using PF Shaping ............................................. 110
4.5. SOLVING THE DIPOLE EQUATIONS ................................................................................................... 111
4.5.1. Solution by Fixing the Free Dipole ......................................................................................... 113
4.5.2. Solution by Utilizing the Free Dipole for Angular Momentum Management ......................... 114
4.6. SUMMARY ....................................................................................................................................... 116
CHAPTER 5
OPTIMAL TRAJECTORY GENERATION FOR EM FORMATIONS............... 119
5.1. INTRODUCTION ................................................................................................................................ 119
5.1.1. An Illustrative Example Using the Gen-X Mission.................................................................. 119
5.2. OPTIMAL CONTROL PROBLEM FORMULATION FOR EMFF............................................................... 124
5.3. OVERVIEW OF SOLUTION METHODOLOGIES .................................................................................... 126
5.4. DIRECT SHOOTING METHOD ............................................................................................................ 128
5.4.1. A Description of the Algorithm ............................................................................................... 128
5.4.2. Some Theoretical Considerations and Limitations of the Algorithm ...................................... 132
5.4.3. Results ..................................................................................................................................... 134
5.5. LEGENDRE PSEUDOSPECTRAL METHOD .......................................................................................... 135
5.5.1. Approximating Functions, their Derivatives and Integrals using Orthogonal Polynomials... 136
5.5.2. Review of the Legendre Pseudospectral Method..................................................................... 143
5.5.3. Results ..................................................................................................................................... 150
5.6. SUMMARY ....................................................................................................................................... 152
CHAPTER 6
REAL-TIME IMPLEMENTATION OF OPTIMAL TRAJECTORIES ............... 155
6.1. INTRODUCTION ................................................................................................................................ 155
6.2. ADAPTIVE TRAJECTORY FOLLOWING FORMULATION ...................................................................... 155
6.2.1. An Upper Bound for Constraint Tightening for Bounded Disturbances ................................. 156
6.2.2. Trajectory Following Algorithm Description.......................................................................... 158
6.3. RECEDING HORIZON CONTROL FORMULATION ............................................................................... 162
6.3.1. RH Formulation with Optimal Cost-to-Go Function .............................................................. 164
6.3.3. Limitations and Possible Extensions....................................................................................... 168
6.4. COMPARISON OF APFM AND OPTIMAL TRAJECTORY FOLLOWING METHOD................................... 169
6.5. SUMMARY ....................................................................................................................................... 172
CHAPTER 7
CONCLUSIONS .......................................................................................................... 175
7.1. SUMMARY AND CONTRIBUTIONS ..................................................................................................... 175
7.2. FUTURE DIRECTIONS ....................................................................................................................... 177
APPENDIX A: MATLAB TOOLS FOR SIMULATION OF ELECTROMAGNETIC
FORMATIONS ........................................................................................................................................ 179
APPENDIX B: PRECISION FORMATION CONTROL USING EMFF .......................................... 183
B.1 GEN-X PRECISION FF REQUIREMENTS............................................................................................. 183
B.2 PROBLEM FORMULATION ................................................................................................................. 184
B.3 REGULATOR SETUP .......................................................................................................................... 186
B.4 PLANT DYNAMICS............................................................................................................................ 189
B.5 DISTURBANCE SOURCES .................................................................................................................. 191
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B.4 CONTROLLER SYNTHESIS ................................................................................................................. 191
B.7 CONCLUSION.................................................................................................................................... 194
REFERENCES ......................................................................................................................................... 197
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List of Figures
Figure 1.1: Equal mass contours for Isp = 3000 thrusters and EMFF for a 10 years mission
life. ............................................................................................................................ 17
Figure 1.2: Thesis organization and interdependency of chapters.................................... 23
Figure 2.1: Geometry of different reference frames ......................................................... 27
Figure 2.2: Geometry of the Rotated Frame Fr................................................................. 34
Figure 2.3: The main magnetic field generated by dynamo action in the hot, liquid outer
core. Above Earth’s surface, nearly dipolar field lines are oriented outwards in the
southern and inwards in the northern hemisphere (source [25])............................... 43
Figure 2.4: Comparison of the tilted dipole model and WMM2005 model for a polar
circular LEO orbit at an altitude of 500 km (the longitude is fixed at -71°)............. 48
Figure 2.5: EMFF coils, reaction wheel and coordinate frame definition for 2D dynamics
................................................................................................................................... 50
Figure 2.6: Geometry of steerable dipoles for magnetic force and torque computation. . 52
Figure 3.1: Two-satellite formation orbits. ....................................................................... 76
Figure 3.2: Follower satellite results................................................................................. 78
Figure 3.3: Follower satellite quaternions. ....................................................................... 78
Figure 3.4: Different estimated parameters ...................................................................... 79
Figure 4.1: A potential function shape for vehicle 1. The obstruction function is at the
location of the vehicle 0. ........................................................................................... 84
Figure 4.2: Reconfiguration results for a two-vehicle formation using APFM................ 86
Figure 4.3: Values of the potential function along the trajectory of vehicle-1. ................ 86
Figure 4.4: The decomposition of the motion planning problem. .................................... 87
Figure 4.5: Case when the gradient of potential function can be zero.............................. 88
Figure 4.6: Coordinate axes for testbed dynamics............................................................ 92
Figure 4.7: A potential function for one satellite in a 4-satellite formation. .................... 96
Figure 4.8: Trajectories for the five-satellite formation reconfiguration........................ 103
Figure 4.9: Lyapunov functions for the five-satellite formation..................................... 103
Figure 4.10: Trajectories for the four-satellite square formation reconfiguration. ......... 104
Figure 4.11: Lyapunov function for the four-satellite square formation. ....................... 105
Figure 4.12: Collision-free trajectories for a 2D, ten-satellite random formation. ......... 106
Figure 4.13: Average simulation and computation-time values for different number of
satellites in the random formations. ........................................................................ 107
Figure 4.14: A potential function for a 2-satellite formation in 2D with potential function
shaping added.......................................................................................................... 109
Figure 4.15: Two-satellite formation reconfiguration result with PF shaping................ 110
Figure 4.16: 2-satellite formation reconfiguration result without PF shaping................ 111
Figure 5.1: Coil architecture for the Gen-X mission with block arrows showing the forces
and torques acting on the dipoles............................................................................ 120
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Figure 5.2: Non-optimal slew results for the Gen-X mission......................................... 123
Figure 5.3: Control profile for the optimal slew maneuver. ........................................... 123
Figure 5.4: Control profile for the optimal slew maneuver. ........................................... 124
Figure 5.5: Relationship between “direct” and “indirect” methods and the Covector
Mapping theorem [47]. ........................................................................................... 127
Figure 5.6: Direct shooting method for solving optimal control problems. ................... 129
Figure 5.7: Results for two-satellite that switch places using the direct shooting method.
................................................................................................................................. 134
Figure 5.8: Cardinal functions (N=5) using Legendre Polynomials and LGL points
(squares) .................................................................................................................. 139
Figure 5.9: Infinity norm of the interpolation error for test functions given by Eq. (5.35)
................................................................................................................................. 142
Figure 5.10: Infinity norm of the derivative for test functions given by Eq. (5.35) ....... 143
Figure 5.11: Optimal trajectory results using the LPS method for a four-satellite
formation 90° CCW rotation................................................................................... 150
Figure 5.12: A ten-satellite random formation optimal trajectories results using LPS
method..................................................................................................................... 151
Figure 5.13: Trajectories generated using LPS method with N=55 for a two-satellite 90°
slew maneuver. ....................................................................................................... 152
Figure 6.1: Trajectory following results for a two-satellite formation using exact BiotSavart force-torque computation for the simulated system model, control and error
signals. .................................................................................................................... 161
Figure 6.2: Trajectory following results for a two-satellite formation using exact BiotSavart force-torque computation for the simulated system model. ........................ 162
Figure 6.3: Receding Horizon formulation using optimal cost-to-go function. ............. 164
Figure 6.4: RH-formulation simulation results............................................................... 167
Figure 6.5: RH-formulation computation time during simulation.................................. 167
Figure 6.6: Comparison of APFM and OTFM trajectories............................................. 170
Figure A.1: Nonlinear 3D Simulink simulation for electromagnetic formation............. 181
Figure B.1: Coordinate setup for control. ....................................................................... 185
Figure B.2: Block diagram for the regulator setup. ........................................................ 186
Figure B.3: Standard State-Space setup for the regulator problem. ............................... 187
Figure B.4: H∞ controller design setup. .......................................................................... 192
Figure B.5: Weighting filters for error W1 and control effort W2. ................................ 193
Figure B.6: Closed-Loop TFs for H∞ synthesized controller.......................................... 194
Figure B.7: Time response of the closed-loop system to disturbances. Ix is total actuator
current, Ic is commanded current and IQ is the current resulting from quantization
process..................................................................................................................... 195
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Chapter 1
Introduction
1.1. Motivation and Background
The term Satellite Formation Flying is used for systems that involve two or more
spacecraft that fly near each other cooperatively to maintain their relative positions and
orientations in order to execute a specific mission. The relative position and orientation of
satellites can be fixed or time varying, e.g., a follower satellite may revolve around the
leader satellite, while this leader-follower formation may itself be in an orbit around
Earth. This distributed satellite architecture in turn enables a number of innovative space
missions that are not possible or infeasible with a larger monolithic satellite structure [1],
[2]. Some authors have defined the term satellite formation flying in a more precise way,
e.g., Scharf et al [3] define satellite formation flying as “a set of more than one spacecraft
whose dynamic states are coupled through a common control law. In particular, at least
one member of the set must 1) track a desired state relative to another member, and 2) the
tracking control law must at the minimum depend upon the state of this other member.”
This definition helps to differentiate the satellite formations from the satellite
constellations (such as GPS) more easily.
Usually there is a very tight requirement for the control of relative position and
orientation of the member satellites in the formation. This in turn may translate into
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constant expenditure of fuel for near-Earth formations or at least fuel is required for
formation reorientation. Therefore, the mission life span depends on the fuel available on
board the satellites in the formation.
In order to elaborate on this point, consider an example of a two-satellite crosstrack formation in a circular orbit around Earth. In such a formation the two satellites
need to be kept at a constant distance from each other in a cross-track direction. It should
be noted that such cross-track formations can be used in Synthetic Aperture Radar (SAR)
applications in which the cross-track distance between the satellites acts as the baseline of
the interferometer and can yield information about ground elevation differences [5].
For a circular orbit, the linearized relative dynamics of each satellite with respect
to the formation center of mass orbit, which is in a natural Keplarian orbit, is given by
Clohessy-Wiltshire equations (also called Hill’s equations) [4]:
x − 2ω0 y = 0
y + 2ω0 x − 3ω02 y = 0
(1.1)
z +ω z = 0
2
0
where the coordinates (x, y, z) are local coordinates defined in a curvilinear reference
frame that rotates as the formation orbits Earth. The y-direction points along the zenith
direction, z-axis points along the orbital plane normal, and x-axis completes the right
hand system (it points along the negative of the velocity vector for a circular orbit). The
formation center of mass mean motion, or orbital frequency, is given by:
ω0 =
μe
R03
(1.2)
where μe is the gravitational constant of Earth (398,600.4418 × 109 m3/s2) and R0 is the
radius of the formation’s center of mass orbit.
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For the cross-track formation, the two satellites are to be kept at a constant
distance d from each other in the z-direction while the separation along the other two axes
is zero. Using Eqs. (1.1), the total inertial force required to hold the formation can be
shown to be:
F =
2ω02 m1m2
d
m1 + m2
(1.3)
where m1 and m2 are the masses of the satellites. The forces are oriented in the zdirection away from the orbital plane, effectively repelling the satellites from each other.
It should be noted that these forces need to be applied constantly to hold the formation
since each satellite in the formation is in a non-Keplerian orbit. Using the rocket equation
[23]:
(
mp = m f e
ΔV / I sp g
)
−1
(1.4)
the mass of propellant required to achieve a certain mission life can be estimated. In this
equation m p and m f are the propellant mass and the final satellite mass (i.e. without
propellant) respectively, ΔV is the “delta V” required to maintain the formation over the
mission lifetime (it can be obtained by integrating Eq. (1.3)), I sp is the specific impulse
of the propulsion system and g is the acceleration due to gravity on the surface of Earth.
Similarly, using the dipole equation for two coaxial coils to generate the force
given by [13]:
2
⎛I ⎞
3
1
F=
μ0 ⎜ c ⎟ ( M 1 R1 )( M 2 R2 ) 4
d
8π ⎝ ρ ⎠
(1.5)
we can estimate the mass of the EMFF subsystem required to maintain the formation. In
this equation F is the force generated between two coaxial coils separated by a distance d,
M 1 R1 and M 2 R2 are the mass and radius product for coils one and two respectively,
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I c / ρ is the ratio of critical coil current and HTS (High Temperature Superconducting)
wire volume mass density and represents a technology factor. Using these two equations
we can plot equal mass contours for EMFF and thrusters, required to hold the formation
in the cross-track direction, as shown in Figure 1.1 (for further detail see Reference [14]).
Figure 1.1 shows the boundary when it is more mass-efficient to use 3000 second Isp
thrusters versus EMFF. The horizontal axis is the cross-track offset (d/2), while the
vertical axis is orbital period. The three curves correspond to different technology levels
of the HTS wire. Three regions of interest exist. To the bottom right, the power needed to
generate the requisite thrust is unavailable and EMFF is the only option. The region
above the curves, and above the thruster limit line, corresponds to cross-track offsets and
orbital periods where thrusters are more mass-efficient. Lastly, in the region above the
thruster-limit line but below the curves, EMFF is more mass-efficient.
This comparison highlights several of the key benefits of EMFF over thrusters as
follows. First, EMFF does not have a mission lifetime limitation. Figure 1.1 shows that
EMFF is better as compared to thruste based systems for LEO and MEO (Medium Earth
Orbits) formations and for long duration missions. Another advantage of EMFF,
highlighted by Eq. (1.5), is that adding an additional satellite to the formation improves
performance of the EMFF subsystem whereas it does not benefit the thruster subsystem
performance. Every time we add an extra, identical satellite and evenly distribute these
satellites in the cross-track direction, the resulting reduction in neighbor-to-neighbor
separation dramatically increases the magnetic force between neighbors. Note the change
in the force given by Eq. (1.5) as a function of distance d. When a third satellite is added
to the center of a two satellite formation, the neighbor-to-neighbor separation is divided
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in half and the neighbor-to-neighbor force increases by a factor of sixteen (24). This
increase in force allows the formation array to be grown in length, a capability
unavailable with thrusters.
Figure 1.1: Equal mass contours for Isp = 3000 thrusters and EMFF for a 10 years mission
life.
Although this example is a rather extreme one in which a steady expenditure of
fuel is required, nevertheless it clearly highlights the impact of consumables on the
mission lifetime. Since the satellites need to carry all of the fuel required over the mission
lifetime (which means they would need to generate higher forces to hold the formation),
using conventional thrusters can become infeasible for certain missions (thruster limit
line in Figure 1.1).
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It should be pointed out that another method of formation flying the satellites,
without the used of consumables, is the concept of tethered formations in which different
satellites in the formation are physically connected together with the help of tethers (see
Reference [6] and references therein for a detailed description of this technique).
1.2. Overview of EMFF and Review of Previous Work on EMFF
The novel concept of Electromagnetic Formation Flight (EMFF) removes the mission life
time dependency on the availability of fuel as highlighted in the last section. EMFF uses
high temperature superconducting (HTS) wire technology to create magnetic dipoles on
each satellite in the formation to generate forces and torques in order to maintain and
reconfigure the satellite formation. A steerable magnetic dipole on each satellite can be
created by using three orthgonogal coils on each satellite in the formation. Force on each
satellite in the formation can be applied in any arbitrary direction by using these steerable
magnetic dipoles. Since these forces are internal, the center of mass of the formation
cannot be moved (momentum is conserved). This can be easily seen for a two satellite
formation in which each satellite experiences equal but opposite force due to magnetic
dipoles on each satellite. Since satellite formation control involves controlling the relative
positions between the satellites, the inability to move the center of mass of the formation
is not a limitation in itself. Another important aspect of EMFF is that whenever a shear
force (i.e., a force that moves a satellite in the lateral direction with respect to the other
satellite) acts on the satellite, a shear torque also acts on it. This shear torque needs to be
countered by angular momentum storage devices such as reaction wheels or control
moment gyros which can also perform the additional task of attitude control. Therefore,
for an arbitrary N-satellite electromagnetic formation, all the relative degrees of freedom
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can be controlled by using three orthogonal coils and three orthogonal reaction wheels.
See references [7], [8] for a more detailed introduction to the concept of EMFF.
Previous work on the dynamics and control problem associated with EMFF has
shown that a two-satellite fully actuated electromagnetic formation that has three
orthogonal coils and three orthogonal reaction wheels is fully controllable [9]. In this
reference the dynamics (including the gyro-stiffening effect) for a two satellite formation
in deep space (i.e. ignoring the gravitational terms, etc.) has also been derived. Models of
the magnetic forces and torques between a general N-satellite electromagnetic formation
are derived (both near-field and far-field) in Reference [10]. This reference also discusses
ways of computing the dipole strengths to achieve the desired forces on the satellites in a
formation. It also discusses the effects of Earth’s gravitational and magnetic fields on
EMFF.
The control of a two satellite formation in LEO (Low Earth Orbit) based on
phase-differences in the coil currents has been proposed by Kaneda et al [11]. In this
method each coil is excited by a sinusoidal current with a constant frequency much
higher than the orbital frequency and the desired force is provided by adjusting the phase
difference between the two dipoles on the satellites in the formation. The sinusoidal
variation in the coil current results in a net cancellation of the magnetic torque acting on
each satellite due to the Earth’s magnetic field. Although this method can be used for a
formation of two satellites, it is not clear how it can be extended to formations of more
than two satellites.
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1.3. Research Objectives and Approach
As discussed in the previous sections, using EMFF allows the mission life to be
independent of the available fuel on board. This comes at a cost in the form of complex
nonlinear and coupled dynamics of the formation, making the control problem more
challenging as compared to conventional thruster based formations. * There are two
factors that increase the complexity of the dynamics and control problem associated with
an electromagnetic formation. Firstly the forces and torques that act on the satellites due
to the magnetic fields of other satellites in the formation are nonlinear, and secondly the
dynamics of each satellite is coupled to that of every other satellite in the formation since
changing the magnetic field strength on one satellite affects every other satellite in the
formation with a non-zero magnetic dipole.
Previous work by Edmond Kong [12], Laila Elias [9] and Samuel Schweighart
[10] has shown that the EMFF concept is feasible for formation flying. This thesis
proposes to take this previous work a step further to find time-optimal trajectories, along
with practical and feasible control schemes to track these trajectories. This will be done
for a general N-vehicle electromagnetic formation in both deep-space and LEO.
In summary the objectives of this thesis are:
•
To develop dynamic models, suitable for the design and analysis of control laws
for electromagnetic satellite formations for both near-Earth and deep space
missions,
*
Another “cost” associated with using EMFF is the thermal control problem on which parallel research is
continuing.
20
•
To develop a framework in which control laws can be designed for
electromagnetic formations for both LEO and deep-space missions,
•
To develop algorithms for the generation of optimal trajectories for a general Nsatellite electromagnetic formation,
•
To develop algorithms for implementing the optimal trajectories in real-time,
•
And to test the models and control laws by using high-fidelity simulations.
The required research to achieve the above mentioned objectives can be broadly
divided into four distinct areas:
1.
Model development,
2.
Control framework,
3.
Optimal trajectory generation and real-time implementation,
4.
Verification in a simulated environment.
Although considerable work has been done in the area of model development,
new models are still needed to meet the objectives. These new models will be developed
using the Lagrangian technique for general electromagnetic formations. Using these
models, the control of general formations will be developed by defining a framework or
architecture in which different control formulations can be designed to meet the particular
mission objectives. Since the “control cost” for EMFF is zero (it uses superconductors
which consume no power and electrical power can be generated abundantly using solar
energy), the optimal trajectories for EMFF are essentially minimum time trajectories
constrained by the system dynamics and control saturations. From a system design point
21
of view, it is important to be able to generate these time-optimal trajectories and also to
actually implement these in real time.
One way of real-time implementation of the optimal trajectories for complex
nonlinear systems is to use the Optimal Trajectory Following Method (OTFM). Since the
algorithms to generate optimal trajectories, for coupled nonlinear and constrained
systems like EMFF, are computationally extensive. Therefore, from a real-time
implementation point-of-view, optimal trajectories will be generated offline and then
followed by separate trajectory following algorithms in real-time (see Chapters 5 and 6
for further detail). Another method of generating the trajectories, with a much lower
computational burden as compared to OTFM, is to use feedback motion planning with
the Artificial Potential Function Method (APFM) (see Chapter 4 for more detail). Both of
these methods will be applied to general electromagnetic formations and the results will
be compared.
1.4. Thesis Overview
The rest of the thesis can be divided into two distinct parts as shown in Figure 1.2. Part I
consists of Chapters 2 and 3 that describe the dynamics and control for general
formations with a special emphasis on LEO. Chapter 2 derives the dynamics of general
Electromagnetic (EM) formations using the Lagrangian method. These dynamics are
suitable for simulating and designing control laws for general N-satellite EM formations
in both LEO and deep-space. Chapter 2 also presents Earth’s magnetic field models.
Chapter 3 presents a framework in which different control laws can be developed for
general EM formations. Using this framework, nonlinear adaptive control laws are
derived in Chapter 3 for formation hold and trajectory following. These adaptive control
22
laws are suitable for both LEO and deep-space applications. This chapter also presents a
practical method of managing angular momentum for general EM formations in LEO.
This is done by switching the direction of the dipoles by 180 resulting in net cancellation
of the torque due to the Earth’s magnetic field. This method is termed dipole polarity
switching control law.
Part I
Part II
Dynamics and Adaptive
Control
Trajectory Generation
Chapter 2
Dynamics
Chapter 4
FB Motion planning using APFM
Chapter 3
Adap. Cont. (LEO)
Dipole polarity switching
Chapter 5, 6
Optimal Traj. Generation and
following
Figure 1.2: Thesis organization and interdependency of chapters.
Part II of the thesis consists of Chapters 4, 5 and 6 that discuss two different
techniques of trajectory generation for nonlinear and constrained systems with specific
application to general EM formations. Chapter 4 presents the Artificial Potential Function
Method (APFM) for generating collision-free trajectories for general EM Formations.
Chapters 5 and 6 present algorithms for implementing optimal trajectories for EMFF in
real time. Chapter 5 presents the formulation of the optimal control problem for general
EM formations. The solution to the optimal control problem is discussed using two
“direct method” algorithms, namely Direct Shooting (DS) and the Legendre
23
PseudoSpectral method (LPS). Chapter 6 discusses two formulations, namely adaptive
trajectory following and Receding Horizon (RH) (for implementing these optimal
trajectories in real time). Thus Chapters 5 and 6 present a practical method, called the
Optimal Trajectory Following Method (OTFM), of implementing optimal trajectories in
real-time for coupled nonlinear and constrained systems. A comparison between APFM
and OTFM is also presented in Chapter 6. The last chapter of the thesis presents a
summary, and chapter-wise synopsis of key contributions of the thesis. Lastly some
future directions for the research are discussed.
24
Chapter 2
Electromagnetic Formation Dynamics
The purpose of this chapter is to develop models that are necessary for simulating general
electromagnetic formations in LEO as well as in deep space. These models will also be
used to design control laws that will be tested in a fully nonlinear simulation
environment. First the dynamic equations of a general electromagnetic formation in LEO
will be derived, which will act as the nonlinear verification model. Next, attitude
dynamics of the satellites in the formation will be presented. Then, deep-space 2D
dynamics will be derived that are suitable for simulating formations for deep space
missions. After that, the Earth’s magnetic field model will be discussed; this is an
essential element for simulating EM formations in LEO. The control laws need to be
designed based on relative dynamics of the formation with respect to a leader satellite.
These relative dynamics will be derived next and the tools required to simulate a general
closed-loop formation in Matlab™ are discussed in Appendix A.
2.1. Translational Dynamic Model for LEO
In this section, nonlinear equations of translational motion for general electromagnetic
satellite formations will be derived in the ECI (Earth Centered Inertial) frame. The use of
the ECI frame results in simpler equations as opposed to using an orbital frame (that is
colocated with the body of the satellite, also sometimes called Hill’s frame). The
25
nonlinear dynamics derived here are useful for simulating a general satellite formation
orbiting Earth. Moreover, for control purposes, these equations can be easily converted to
relative equations of motion as presented later in this chapter.
2.1.1. Preliminaries
As discussed above, the equations of motion are developed in the ECI reference frame,
which has its origin at the center of Earth, its x-axis points towards vernal equinox *, zaxis towards celestial north pole, and y-axis completes a right handed axis system. This
axis system is not fixed to Earth (i.e., this frame does not rotate with Earth), although it
moves as Earth orbits around the sun. For the purposes of this thesis, this frame can be
assumed to be an inertial frame. Moreover, relative positional dynamics will be
developed in an orbital frame FRO. This orbital frame is defined in such a way that its
origin is attached to the center of mass of the formation with its y-axis aligned with the
position vector R RO representing the position of the formation center of mass in the ECI
frame. The z-axis points towards the orbital plane normal and the x-axis completes the
right hand system (see Figure 2.1). Also a body frame, FBk, attached to the body of
satellite-k with its origin at the center of mass of the satellite, is also defined which
defines the orientation of each satellite in the inertial space.
In order to emphasize the peculiarities of electromagnetic formation control, we
will restrict the dynamics to a circular orbit in order to avoid unnecessary details,
although the same procedure can be used to extend the dynamics to elliptical orbits. The
dynamics are derived for a general N-satellite electromagnetic formation such that the
*
Vernal equinox is the direction of intersection of the Earth’s equatorial plane and the plane of the Earth’s
orbit around the sun (ecliptic) when the sun crosses the equator from south to north in its apparent annual
motion along the ecliptic [15].
26
satellites are enumerated from 0 to N-1, with satellite-0 designated as the “leader”
satellite.
Satellite j
Satellite i
rij
FBi
FBj
Reference Orbit
Ri
r0j
r0i
Formation c.m.
RO
F
Rj
“Leader”
RRO
FB0
Earth
R0
FI
Figure 2.1: Geometry of different reference frames
For the derivation of the relative dynamics in the orbital frame it is necessary to
represent the rate of change of a vector in two different reference frames. This can be
achieved by using the “Transport Theorem” [19]:
FI
r=
FRO
r+
FRO
where FI denotes the ECI reference frame,
FRO in frame FI,
FRO
FI
FI
ω ×r
FRO
(2.1)
FI
ω denotes the angular velocity of frame
r is the velocity as observed by an observer attached to frame FI, and
r is the velocity observed by an observer attached to frame FRO and, × denotes cross
product.
27
2.1.2. Derivation of Translational Equations of Motion
Let Ri be the position vector of the ith satellite in a general N-satellite formation. Note
that R i is a vector (a geometric object) that can have many representations in a given
reference frame, for example a suitable representation for R i in the ECI reference frame
is:
R i = xi ax + yi a y + zi az
(2.2)
where:
ax = Unit vector that points in the direction of the vernal equinox
az = Unit vector that is aligned with the celestial North Pole
a y = az × ax
(Note that a general vector R i is denoted in a given reference frame as Ri in the
following discussion).
The equations of motion will be developed using Lagrange’s method [18], which
is particularly useful for incorporating the gravitational perturbation terms and the
magnetic force actuation terms. The velocity of the ith satellite, modeled as a point mass,
is Ri and its kinetic energy is given as:
Ti = 12 mi RiT Ri
where mi is the mass of the satellite and Ri = [ xi
(2.3)
yi
zi ]T . The potential energy of the
satellite in Earth’s gravitational field is given as [15]:
k
⎤
⎛ Re ⎞
⎢1 − ∑ J k ⎜ ⎟ Pk (cos φ ) ⎥
Vi ( Ri , φ ) = −
Ri ⎢ k = 2 ⎝ Ri ⎠
⎥⎦
⎣
g
μe mi ⎡
∞
where:
28
(2.4)
μe = Gravitational constant of Earth = 398,600.4418 × 109 m3/s2
Ri = xi 2 + yi2 + zi2
Re = Equatorial Radius of Earth = 6378136.49 m
Jk = kth zonal harmonic of Earth (J2 = 0.00108263, J3 = -0.00000254, J4 = 0.000000161)
Pk = Legendre polynomials of the first kind
φ = Angle between Earth’s North Pole direction, i.e. az and Ri
Note that Eq. (2.4) accounts for the non-spherical nature of the Earth (i.e. the
equatorial radius is larger than the polar radius) and assumes that Earth is symmetrical
about its rotation axis hence other effects which are called tesseral and sectorial
harmonics [16] are ignored. Although these effects may also be included in this
expression, the J2 term is by far the dominant term and only this term will be included in
further analysis in this thesis for simplicity (although the Lagrange’s method presented
here allows for the addition of other terms in a straight forward manner). Equation (2.4),
neglecting J3 and higher terms, gives the gravitational potential energy of the ith satellite
as:
2
2
⎤
⎛ Re ⎞ ⎛ zi ⎞
⎢1 − J 2 ⎜ ⎟ (3 ⎜ ⎟ − 1) ⎥
Vi ( Ri ) = −
Ri ⎢
⎥⎦
⎝ Ri ⎠ ⎝ Ri ⎠
⎣
g
μe mi ⎡
(2.5)
where we have used the fact that cos(φ ) = zi / Ri in the ECI reference frame.
In a similar fashion each satellite is immersed in the magnetic field generated by
other satellites, which results in magnetic forces acting on the satellite. To incorporate the
effect of these magnetic fields we need to determine the magnetic potential energy of a
satellite due to the magnetic field of the other satellites in the formation. Modeling the
29
coils as magnetic dipoles, the magnetic potential energy of the ith satellite due to the
magnetic field of the jth satellite is given as [17]:
Vijm ( Ri , R j ) = − μi i Bij (rij )
(2.6)
where μi is the dipole strength of the ith satellite (which is a vector quantity since we
have assumed three orthogonal coils on each satellite hence giving us the ability to
control current in each coil to orient this dipole vector anywhere in Euclidean space
3
),
rij is a vector that gives the position of the jth satellite with respect to the ith satellite, B ij
is the magnetic field strength due to satellite j at the location of the ith satellite, and "i" is
the dot-product operator. Note that the “dipole model” of the coils on the satellites is
essentially a far-field approximation of the magnetic field and gives accurate results only
when the distance between the satellites is many times the radius of the coils. See
Reference [10] for a detailed derivation of the far-field model of the magnetic field. Since
the principle of superposition applies to the magnetic field, to determine the total
magnetic potential energy of the ith satellite due to the magnetic field of all the other
satellites in the formation we simply sum up the individual contributions:
⎛ N −1
⎞
Vi m ( R0 , R1 ,..., RN −1 ) = − μi i⎜ ∑ Bij (rij ) ⎟
⎝ j =0, j ≠i
⎠
(2.7)
Equation (2.7) will be developed further a bit later in this section due to the complexity of
the resulting expressions.
Similarly the ith satellite also experiences the Earth’s magnetic field and its
magnetic potential energy due to Earth’s magnetic field is given as:
Vi mE ( Ri ) = − μi i Be ( Ri )
where Be is the Earth’s magnetic field vector at the location of the ith satellite.
30
(2.8)
The equations of motion of the formation of N satellites can be derived using the
Lagrange’s equations [19]:
d
( Lq ) − Lq = Q
dt
(2.9)
where:
L(q, q ) = System Lagrangian =
N −1
N −1
i =0
i =0
{
}
∑ Ti (qi ) − ∑ Vi g (qi ) + Vi m (q) + Vi mE (qi ) ,
q = Vector of generalized coordinates of the system = [q0 q1 q2 … qN-1]T,
qi = Generalized coordinates of the ith satellite = [xi yi zi]T,
Q = Vector of generalized forces acting on the formation (such as solar pressure,
drag, etc),
Lq =
∂L
∂L
, and Lq =
∂q
∂q
From the expression of the system Lagrangian given above, it can be argued that
the equations of motion of the ith satellite can be developed by considering only the
Lagrangian associated with it, i.e.:
d i
Lq − Liqi = Qi
dt i
( )
(2.10)
This is possible since the kinetic energy of the ith satellite depends only on its velocity
and not on the generalized coordinates of any other satellite in the formation (as we are
deriving the equations in an inertial reference frame). Carrying out the gradient and time
derivative operations in Eq. (2.10) we get:
31
mi xi −
∂
Vi g + Vi m + Vi mE ) = Qix
(
∂xi
mi yi −
∂
Vi g + Vi m + Vi mE ) = Qiy
(
∂yi
mi zi −
∂
(Vi g + Vi m + Vi mE ) = Qiz
∂zi
(2.11)
where the mass of the satellite, mi, is assumed to be constant and the Qi’s represent the
components of the external disturbance forces acting on the ith satellite in the ECI
reference frame. Using Eq. (2.5), the gradient of the gravitational potential in the ECI
frame can be expressed as:
⎡ xi ⎤
μe mi ⎢ ⎥ 3μe Re2 J 2 mi
∂Vi
= − 3 ⎢ yi ⎥ +
2 Ri7
Ri
∂qi
⎢⎣ zi ⎥⎦
g
⎡ xi (5 zi2 − Ri2 ) ⎤
⎢
2
2 ⎥
⎢ yi (5 zi − Ri ) ⎥
⎢ zi (5 zi2 − 3Ri2 ) ⎥
⎣
⎦
(2.12)
Using Eq. (2.7), the gradient of the magnetic potential can be written as:
⎞
∂
∂ ⎛ N −1
⎡⎣Vi m (q1 , q2 ,..., qN ) ⎤⎦ = − μi i
⎜ ∑ B j (rij ) ⎟ =
∂qi
∂qi ⎝ j =0, j ≠i
⎠
N −1
∑
j = 0, j ≠ i
F (qi , q j , μi , μ j ) =
m
ij
FI
N −1
∑
j = 0, j ≠ i
Fijm (qi , q j , μi , μ j )
(2.13)
Fi (q1 ,..., qN , μ1 ,..., μ N )
m
where Fijm is the magnetic force that acts on satellite-i due to satellite-j, rij = q j − qi , and
FI
Fi m is the net magnetic force acting on satellite-i due to all other satellites in the
formation (the pre-superscript FI is added to emphasize that this force is in the ECI
frame). Note that since the magnetic force is a conservative force, i.e. the change in the
magnetic potential energy of a current carrying coil moving through the magnetic field is
independent of the path taken by the coil, hence it can be written as the gradient of the
magnetic potential. To see that this is the case, for a closed path the net work done is zero
in the magnetic field:
32
∫ F.dr = 0
and using Stoke’s theorem in
3
(2.14)
[20], we can write the circulation integral of Eq. (2.14)
as a curl (which is circulation per unit area at a point) as follows:
∇×F = 0
(2.15)
∇ × (∇f ) = 0
(2.16)
Using the vector calculus identity:
for a scalar field f ( x, y, z ) in the Euclidean space
m
F ij (qi , q j , μi , μ j ) = ∇Vijm = − μi i
3
, the force Fijm can be written as:
∂ B j (rij )
(2.17)
∂qi
Using the dipole approximation of the coils on each satellite, the magnetic field
due to the jth satellite at the location of the ith satellite can be written as [9]:
B j (rij ) =
μ ⎞
μ0 ⎛ 3μ j irij
rij − 3j ⎟
⎜
5
rij ⎠
4π ⎝ rij
(2.18)
Taking the gradient of Eq. (2.18), Eq. (2.17) can be written as:
m
F ij (qi , q j , μi , μ j ) =
(
)(
)
⎞
μ ir μ ir
3μ0 ⎛ μi i μ j
μ ir
μ ir
⎜ − 5 rij − i 5 ij μ j − j 5 ij μi + 5 i ij 7 j ij rij ⎟
⎟
4π ⎜⎝
rij
rij
rij
rij
⎠
(2.19)
Note that Eq. (2.19) gives the force on dipole i (present on satellite-i) due to
dipole j (located on satellite-j). It depends on the distance between the two dipoles and
the orientation of both dipoles in the inertial space. It is the dependence on the orientation
of the dipoles that gives rise to the complexity of the expression for the force since the
orientation of a dipole depends obviously on its orientation with respect to the body
33
frame of the satellite; it also depends on the orientation of the body axes in the inertial
space. A simple algebraic form of the force equation can be obtained by defining a
rotated frame, Fr, such that its x-axis is aligned with vector rij (see Figure 2.2).
Figure 2.2: Geometry of the Rotated Frame Fr.
In this rotated frame Fr, Eq. (2.19) can be expressed as:
⎡ 2 μix μ jx − μiy μ jy − μiz μ jz ⎤
3
μ
⎥
Fr m
0 ⎢
− μix μ jy − μiy μ jx
Fij (qi , q j , μi , μ j ) =
4 ⎢
⎥
4π rij
⎢
⎥
μ
μ
μ
μ
−
−
ix jz
iz jx
⎣
⎦
(2.20)
where pre-superscript Fr is added to emphasize that this force is in frame Fr, and rij is the
distance between the two dipoles and individual dipole strengths are expressed in their
Cartesian components, i.e. μk = [ μkx
μky
μkz ]T ( k ∈ {i, j} ) in the rotated frame Fr. Note
that in Eq. (2.20), the dipole components of satellite-i and satellite-j depend on their
34
respective orientations, hence the dependence of the force equations on the attitude of
each satellite is implicitly embedded in the vector notation for μi and μ j .
Equation (2.20) highlights the two nonlinearities associated with using magnetic
dipoles as actuators. The first of these nonlinearities is associated with the magnitude of
the force, which scales inversely with the distance to the fourth power. The second of
these nonlinearities is associated with both the magnitude and direction of the force as
these depend upon the product of the individual components of the two dipoles. This
second nonlinearity highlights the coupled nature of the magnetic actuation, i.e.,
modifying one dipole on one of the satellites in the formation results in the change of
force on every other satellite in the formation with non-zero magnetic dipole.
Since we need the magnetic forces between the satellites in the ECI reference
frame, we can use Eq. (2.19) directly or we can use Eq. (2.20) after multiplying it with
the transformation matrix as follows:
FI
where
FI
Fijm (qi , q j , μi , μ j ) = FI T Fr Fr Fijm (qi , q j , μi , μ j )
(2.21)
T Fr is the orthogonal transformation matrix from the rotated frame to the inertial
ECI reference frame. Such a matrix can be constructed since we need two principal
rotations of the ECI reference frame to align it with the rotated frame Fr. We first rotate
the ECI reference frame about its z-axis by an angle ψ and then the resulting frame is
rotated about its y-axis by an angle θ to complete the orientation to frame Fr. Thus the
transformation matrix from the ECI to the Fr frame can be written as [21]:
Fr
T
FI
⎡cos θ
= ⎢⎢ 0
⎢⎣ sin θ
0 − sin θ ⎤ ⎡ cosψ
1
0 ⎥⎥ ⎢⎢ − sinψ
0 cos θ ⎥⎦ ⎢⎣ 0
35
sinψ
cosψ
0
0⎤
0 ⎥⎥
1 ⎥⎦
(2.22)
The inverse transformation from the Fr frame to the ECI reference frame FI is given
simply by the inverse of the matrix
Fr
T FI (which is equal to its transpose). The angles ψ
and θ can be determined from the vector rij as follows:
⎛ rij − x
⎜r
⎝ ij − y
⎞
⎟⎟
⎠
ψ = tan −1 ⎜
⎛r
θ = sin ⎜ ij − z
⎜ rij
⎝
−1
⎞
⎟
⎟
⎠
(2.23)
where rij − x , rij − y , and rij − z are the x, y and z-components, respectively, of the vector rij in
the ECI reference frame. Moreover, the magnetic dipoles of each satellite are needed in
the ECI reference frame to use Eq. (2.19). For that we need the transformation matrices
from the body frame of each satellite to the global ECI reference frame, which are
described in the next section where the attitude dynamics will be discussed.
In a similar fashion, the gradient of the Earth’s magnetic potential gives the force
on the ith satellite due to the Earth’s magnetic field. It will be shown later in this chapter
(when Earth’s magnetic field model will be discussed) that this gradient is extremely
small as compared to other forces present in the system and can safely be ignored for the
purposes of this thesis.
After combining these results with Eqs. (2.11), we obtain the translational
dynamics of the ith satellite in the ECI reference frame as follows:
mi xi +
mi yi +
mi zi +
μe mi xi
Ri3
μe yi mi
3
i
R
μe zi mi
3
i
R
+
3μe Re2 J 2 mi 2
( Ri − 5 zi2 ) xi =
2 Ri7
+
3μe Re2 J 2 mi 2
( Ri − 5 zi2 ) yi =
7
2 Ri
FI
3μe Re2 J 2 mi
+
(3Ri2 − 5 zi2 ) zi =
7
2 Ri
Fixm (q1 ,..., qN , μ1 ,..., μ N ) + FI Fdix
FI
Fiym (q1 ,..., qN , μ1 ,..., μ N ) + FI Fdiy
FI
36
Fizm (q1 ,..., qN , μ1 ,..., μ N ) + FI Fdiz
(2.24)
In these equations the notation has been changed from Q to Fd to emphasize that
these are disturbance forces. Also note that Eqs. (2.24) include the J2 perturbations and
describe the translational dynamics of the ith satellite in the ECI reference frame. These
equations, along with Eq. (2.19), can be used to simulate a general electromagnetic
formation around Earth in the ECI frame and act as a nonlinear verification model for
validation and testing of control laws developed in a later chapter. Moreover, these
equations can be used to find the relative orbital dynamics of the satellites as discussed
later in this chapter.
2.2. Attitude Dynamics Including Gyro-Stiffening Effects
As discussed previously a fully actuated satellite that has three orthogonal coils can
control all the relative translational degrees of freedom. One side effect of applying any
shear force using magnetic dipoles is that a torque also acts on the dipoles; therefore to
control the attitude of a satellite and to counter this magnetic torque, each satellite in the
formation needs to have angular momentum storage devices such as reaction wheels or
Control Moment Gyros (CMGs). In this section, the attitude dynamics of a satellite in an
electromagnetic formation will be presented in a simple form in order to highlight the
control issues associated with attitude control. For example, it is assumed that the
reaction wheels are mounted rigidly on the satellite (see [9] for modeling details for
flexible mountings of the reaction wheels). It will be assumed that each satellite has three
orthogonal reaction wheels and their gyro-stiffening effect will also be included in the
dynamic equations.
For a rigid body rotating about its center of mass, the rate of change of the angular
momentum is related to the applied torques and is given by the Euler equation [21]:
37
FI
(2.25)
H =τ
where H is the total angular momentum of the satellite (including that of the reaction
wheels), due to its rotation about its center of mass, and τ is the external torque acting on
the satellite. Using the Transport Theorem, Eq. (2.1), Eq. (2.25) can be written in terms of
the rate of change of angular momentum in the body frame of the satellite as:
H +ω × H =τ
(2.26)
where ω is the angular velocity of the satellite in the inertial frame. The angular
momentum of the satellite can be written as the sum of satellite body momentum (without
reaction wheels spinning) and the reaction wheel momentum:
H = H B + H RW
(2.27)
Using Eq. (2.27) the attitude dynamics can be written as:
m
H B + H RW + ω × H B + ω × H RW = τ + τ
where τ
m
d
(2.28)
is the magnetic torque acting on the satellite due to the magnetic field of other
satellites in the formation and τ
d
is the disturbance torque (such as due to Earth’s
magnetic field). Moreover the rate of change of angular momentum of the reaction
wheels acts as a torque on the satellite body frame, therefore we can write Eq. (2.28) as:
m
d
H B + ω × H B + ω × H RW = τ + τ − τ RW
(2.29)
Let I and IRW be the inertia matrices of the satellite and reaction wheels,
respectively; then since H B = I ω and H RW = I RW Ω (where Ω is the reaction wheel’s
angular velocity vector), under the assumptions that the satellite body axes are aligned
with the principal axes and the reaction wheels are also aligned with the satellite body
axes, Eq. (2.29) reduces to following attitude dynamic equations:
38
I1ω1 + ( I 3 − I 2 )ω2ω3 + ω2 I RW 3Ω3 − ω3 I RW 2Ω 2 = τ mx + τ dx − τ RWx
I 2ω2 + ( I1 − I 3 )ω1ω3 + ω3 I RW 1Ω1 − ω1 I RW 3Ω3 = τ my + τ dy − τ RWy
(2.30)
I 3ω3 + ( I 2 − I1 )ω1ω2 + ω1 I RW 2 Ω 2 − ω2 I RW 1Ω1 = τ mz + τ dz − τ RWz
where ω = [ω1 ω2 ω3 ]T is the angular velocity of the satellite in the ECI reference
frame, and
⎡ I1
I = ⎢⎢ 0
⎢⎣ 0
I RW
0
I2
0
⎡ I RW 1
= ⎢⎢ 0
⎢⎣ 0
0⎤
0 ⎥⎥ = the inertia matrix for the satellite,
I 3 ⎥⎦
0
I RW 2
0
0 ⎤
0 ⎥⎥
I RW 3 ⎥⎦
= the inertia matrix for the reaction wheels,
τ m = [τ mx τ my τ mz ]T , τ d = [τ dx τ dy τ dz ]T , and τ RW = [τ RWx τ RWy τ RWz ]T =
torques expressed in the satellite body frame (aligned with principal axes).
Note that Eqs. (2.30) are the simplest possible form of attitude dynamics for an
EMFF satellite with three orthogonal reaction wheels which are aligned with the principal
axes of the satellite. The orientation of the satellite in the inertial frame can be related to
the body axis angular rates through either Euler angles or quaternions as follows [22]:
1
q = − Q(q )ω
2
(2.31)
θ = M (θ )ω
(2.32)
where:
q = [q0
q1
q2
T
q3 ] = a quaternion vector,
θ = [θ1 θ 2 θ3 ]T = Euler angle vector,
and the matrices Q and M are defined by
39
⎡ q1
⎢ −q
=
Q(q) ⎢ 0
⎢ − q3
⎢
⎣ q2
q3 ⎤
−q2 ⎥⎥
,
q1 ⎥
⎥
− q0 ⎦
q2
q3
− q0
− q1
⎡1 sin θ1 tan θ 2
M (θ ) = ⎢⎢0
cos θ1
⎢⎣0 sin θ1 sec θ 2
cos θ1 tan θ 2 ⎤
− sin θ1 ⎥⎥ .
cos θ1 sec θ 2 ⎥⎦
Note that by using quaternions we can avoid the singularities present in using
Euler angles, but if there are no large angle slews then Euler angles can be used as well,
while avoiding the singularities.
2.2.1. Magnetic Torque
The magnetic torque term in Eq. (2.28) can be written as:
m
τi =
N −1
∑
m
τ ij
(2.33)
j = 0, j ≠ i
m
where τ ij is the magnetic torque acting on the satellite-i dipole due to the satellite-j
dipole. This torque can be written as:
m
τ ij = μi × B j (rij )
(2.34)
where the terms are as defined in Eq. (2.17). Again, as we did in the case of magnetic
force, using the dipole approximation of the coils on each satellite, the magnetic torque
on the ith satellite due to the jth satellite can be written using Eq. (2.18) as follows:
m
τ ij =
μ0 μi ⎛ 3rij ( μ j irij ) μ j
×⎜
− 3
4π ⎜⎝
rij5
rij
⎞
⎟
⎟
⎠
(2.35)
Again the expression for torque is complicated by the fact that torque depends upon the
orientation of both the source and target dipoles. As in the case of magnetic force, a
40
simple expression for magnetic torque can be derived in a rotated frame Fr as defined for
Eq. (2.20):
Fr
⎡ − μiy μ jz + μiz μ jy ⎤
μ0 ⎢
⎥
τ =
2 μiz μ jx + μix μ jz ⎥
3 ⎢
4π rij
⎢ − μix μ jy − 2 μiy μ jx ⎥
⎣
⎦
m
ij
(2.36)
Equation (2.36) highlights the nonlinearities associated with magnetic torque,
which were also present with magnetic force, but with one important difference, namely
the magnitude of the magnetic torque is inversely proportional to distance to the 3rd
power (as opposed to the force that depends on the distance to the 4th power). Therefore,
the same magnetic field can produce much stronger torque, as compared to force, in a
given dipole (the reason that magnetic torquers are used for satellite attitude control [23])
although for EMFF this may act as a negative factor during the system design process.
In order to compute the magnetic torque using Eq. (2.35), we need to express the
magnetic dipoles μi and μ j in the ECI reference frame as is the case for computing the
force using Eq. (2.19). For that we need transformation matrices for each satellite to
transform a vector from the body frame FBi to the inertial frame FI and vice versa. This
can be achieved using the Euler angles as follows:
FBi
where
T FI
FBi
0
⎡1
⎢
= ⎢ 0 cos θ 3
⎢⎣ 0 − sin θ 3
0 ⎤ ⎡ cos θ 2
sin θ 3 ⎥⎥ ⎢⎢ 0
cos θ 3 ⎥⎦ ⎢⎣ sin θ 2
0 − sin θ 2 ⎤ ⎡ cos θ1
1
0 ⎥⎥ ⎢⎢ − sin θ1
0 cos θ 2 ⎥⎦ ⎢⎣ 0
sin θ1
cos θ1
0
0⎤
0 ⎥⎥
1 ⎥⎦
(2.37)
T FI is the transformation matrix from the ECI frame FI to the satellite-i body
frame FBi. Note that this is just one of the possible twelve ways in which the
transformation matrix can be defined using Euler angles and this form is most common in
aerospace applications (see reference [21] page 20). The transformation defined by Eq.
(2.37) corresponds to first rotating the inertial frame about its z-axis by the angle θ1 , then
41
the resulting intermediate frame is rotated about its y-axis by the angle θ 2 , and finally
this second resulting intermediate frame is rotated about its x-axis by the angle θ3 .
Moreover the inverse transformation, i.e., the transformation of a vector from the
satellite-i body frame to the inertial frame is the inverse of the above matrix which is
simply given by the transpose of the matrix since it is orthonormal:
FI
T FBi = ( FBi T FI )
T
(2.38)
It is a bit more efficient, from a computational point of view, to use quaternions to
define the rotation matrix. The FBi to FI transformation matrix in terms of quaternions can
be expressed as [22]:
FI
T FBi
⎡ 2q02 + 2q12 − 1 2q1q2 + 2q0 q3 2q1q3 − 2q0 q2 ⎤
⎢
⎥
= ⎢ 2q1q2 − 2q0 q3 2q02 + 2q22 − 1 2q2 q3 + 2q0 q1 ⎥
⎢ 2q1q3 + 2q0 q2 2q2 q3 − 2q0 q1 2q02 + 2q32 − 1 ⎥
⎣
⎦
(2.39)
and the inverse transformation is given by Eq. (2.38).
2.3. Earth’s Magnetic Field Model
A very important factor present in the disturbance torque term in Eq. (2.28) is the torque
that acts on the satellite due to the Earth’s magnetic field. This torque can be written as:
e
τ i = μ i × B e ( Ri )
(2.40)
where B e is Earth’s magnetic field strength at the location of the satellite. It is a vector
quantity and varies both with location and time. At orbital altitudes, the magnetic field of
the Earth can be described as a combination of two components as follows:
B e ( Ri , t ) = B m ( Ri , t ) + B d ( Ri , t )
(2.41)
where B m ( R i , t ) is the main field component, which is due to the core of the planet, and
B d ( R i , t ) is a disturbance component (that can be as much as 10% of the main field at
42
different orbital locations), which is due to sources such as currents in the ionosphere and
solar wind effects on the Earth’s magnetosphere [24]. Note that Eq. (2.41) includes time
as an independent variable to emphasize that the magnetic field varies slowly over time.
At the surface of the Earth, the main field component has two sources, namely due to the
core of the planet and magnetized rocks near the surface of the Earth. But at orbital
altitudes, we can ignore the effect of magnetized rocks.
2.3.1. Tilted Dipole Model of the Earth’s Magnetic Field Main Component
The main field of Earth (see Figure 2.3) can be modeled as a tilted dipole located at the
center of the earth. The orientation of this dipole varies slowly over time and the current
magnetic North pole location is at 71.78º W and 79.74º N (2005 estimate) [25].
Figure 2.3: The main magnetic field generated by dynamo action in the hot, liquid outer
core. Above Earth’s surface, nearly dipolar field lines are oriented outwards in the
southern and inwards in the northern hemisphere (source [25])
By defining a geomagnetic reference frame Fm, the magnetic field at the location
of satellite i can be computed using Eq. (2.18) as follows. Let Rim = [ xi
43
yi
zi ]T be the
location of the ith satellite in the magnetic reference frame Fm. Since, in the magnetic
frame, the Earth’s magnetic dipole is oriented along the z-axis (see Figure 2.3), i.e.
μe = [0 0 − μm ]T where μm is the dipole strength of the Earth’s magnetic field which
approximately equals 8.0×1022 A-m2 (see page 14 of reference [26]), we can expand Eq.
(2.18) as follows:
Fm
⎡ xi zi
⎤
3μ 0 μ m ⎢
Be ( Ri ) = −
yi zi ⎥⎥
5 ⎢
4π Rim 2
⎢⎣ zi − Rim2 / 3⎥⎦
(2.42)
or, in terms of the latitude and longitude of the satellite in the magnetic reference frame,
we have:
xi = Rim cos λm cosη m
yi = Rim cos λm sin ηm
(2.43)
zi = Rim sin λm
where η m is the magnetic longitude and λm is the latitude with respect to the
geomagnetic equatorial plane. Using the above equations, Eq. (2.42) can be written as:
Fm
⎡3sin λm cos λm cosη m ⎤
μ0 μ m ⎢
3sin λm cos λm sin η m ⎥⎥
Be ( Ri ) = −
3 ⎢
4π Ri
⎢⎣
⎥⎦
3sin 2 λm − 1
(2.44)
The magnetic torque on satellite i can be approximated by using Eqs. (2.40) and
(2.44). Note that in order to use Eq. (2.42) or Eq. (2.44), the location of the satellite must
be expressed in the frame Fm. This can be done by first transforming the coordinates of
the satellite in the ECI frame to the ECEF (Earth Centered Earth Fixed) reference frame
by using the following rotation matrix:
FE
T
FI
⎡ cos φe
= ⎢⎢ − sin φe
⎢⎣ 0
sin φe
cos φe
0
44
0⎤
0 ⎥⎥
1 ⎥⎦
(2.45)
where φe is the rotation angle of Earth in the ECI reference frame. This angle can be
computed using the following relation:
φe = φ0 + ωet
(2.46)
where:
φ0 = Greenwich longitude w.r.t. vernal equinox (x-axis of ECI) at t = 0
ωe = Rotation rate of Earth = 7.2722 × 10-5 rad/s
t = Simulation time
The transformation matrix from ECEF to the magnetic frame Fm is given by:
Fm
T
FE
⎡cos λMP
= ⎢⎢ 0
⎢⎣ sin λMP
0 − sin λMP ⎤ ⎡ cos φMP
⎥ ⎢ − sin φ
1
0
MP
⎥⎢
0 cos λMP ⎥⎦ ⎢⎣ 0
sin φMP
cos φMP
0
0⎤
0 ⎥⎥
1 ⎥⎦
(2.47)
where:
λMP = Colatitude of the magnetic North Pole = 10.26° (2005 estimate)
φMP = Longitude of the magnetic North Pole = -71.78° (2005 estimate)
It should be noted that the tilted dipole model of the Earth’s magnetic field is only
an approximation of the main field component and it can have significant errors in LEO,
although this error reduces with increasing altitude [24]. For a more realistic model of the
Earth’s magnetic field, existing geomagnetic models such as IGRF (International
Geomagnetic Reference Field) or WMM (World Magnetic Model) can be used. Since
these models also give only the Earth’s main field components, it is essential that any
control scheme for EMFF must account for the uncertainty present in the magnetic model
by using online estimation or direct measurement.
45
2.3.2. Full Representation of the Main Field using Spherical Harmonic Analysis
Away from the sources (i.e. current) the magnetic field can be computed from the
solution of Laplace’s Equation [17]:
∇ 2Vm = 0
(2.48)
where Vm is the magnetic potential function. The magnetic field itself is given by the
gradient of the potential function, i.e.:
B = −∇Vm
(2.49)
In spherical coordinates, Eq. (2.48) can be written as:
∂Vm ⎞
∂ 2Vm
1 ∂ 2 (rVm )
1
∂ ⎛
1
+
sin
+
=0
θ
∂θ ⎟⎠ r 2 sin 2 θ ∂φ 2
r ∂r 2
r 2 sin θ ∂θ ⎜⎝
(2.50)
where:
θ = colatitude,
φ = longitude,
r = radius.
A solution to the above equation can be obtained by the method of separation of
variables and has the following form [27]:
∞
⎛a⎞
Vm = a ∑ ⎜ ⎟
n =1 ⎝ r ⎠
n +1 n
∑ ⎡⎣ g
m =0
m
n
cos ( mφ ) + hnm sin ( mφ ) ⎤⎦ Pnm (θ )
(2.51)
where g nm and hnm are time dependent Gauss coefficients, Pnm are associated Legendre
polynomials of order m and degree n, and a is the standard magnetic reference radius of
Earth (6371.2 km). Note that this solution assumes that there are no sources external to
Earth, which can be seen from the fact that Vm approaches zero for large r.
The general solution to Laplace’s Equation can be seen as an infinite combination
of terms, where each of these terms satisfies Laplace’s Equation. These terms, namely
46
Pnm (θ ) cos ( mφ ) and Pnm (θ ) sin ( mφ ) are called spherical harmonics and have a nice
property that they are orthogonal to each other (see reference [27] for detail). The
orthogonality of these spherical harmonics enables us to determine the unknown Gauss
coefficients by fitting the model given by Eq. (2.51) to a set of carefully measured data.
The resulting problem is a set of linear equations that can be solved by using least-square
estimation techniques [25]. The resulting set of coefficients is what essentially comprises
the WMM (World Magnetic Model). These coefficients, along with Eqs. (2.51) and
(2.49), can be used to estimate the main component of the Earth’s magnetic field at the
desired location in the ECEF reference frame.
It should be pointed out that the first three terms of the WMM, namely g10 , g11 ,
and h11 , correspond to the tilted dipole model of the Earth’s magnetic field. The strength
of the tilted dipole is related to these first three coefficients as follows:
μm =
4π Re3
μ0
( g ) + ( g ) + (h )
0 2
1
1 2
1
1 2
1
(2.52)
According to WMM2005, the values of the first three coefficients are -29556.8, -1671.7,
and 5079.8, respectively [25]. Using these values gives us the dipole strength of
7.7681×1022 A-m2. Using these coefficients, the colatitude λMP and longitude φMP of the
titled dipole can be computed as follows [26]:
⎛ (h1 ) 2 + ( g 1 ) 2
1
1
λMP = − tan ⎜
⎜
g10
⎝
1
−1 ⎛ h1 ⎞
φMP = tan ⎜ 0 ⎟
⎝ g1 ⎠
−1
⎞
⎟
⎟
⎠
(2.53)
The two models are compared for the Earth’s B-field values for a LEO orbit as
shown in Figure 2.4. As can be seen, there is significant difference between the results of
47
the two models and the tilted model can be off from the WMM2005 by as much as 50%.
Hence it is important that for realistic simulations, the more accurate WMM2005 or
IGRF must be used. But, for the purposes of evaluating the performance of the EMFF
system from a dynamics and control perspective, the tilted dipole model (which is
simpler to simulate) provides enough detail and is sufficient for the purposes of this
thesis.
4
2
x 10
4
Bx (x-comp. of Earth's B-field)
4
x 10
By (y-comp. of Earth's B-field)
WMM2005
Tilted Dipole
2
nT
nT
1
0
-1
0
-2
WMM2005
Tilted Dipole
-2
-100
-50
4
5
x 10
0
latitude (deg)
50
-4
-100
100
-50
4
Bz (z-comp. of Earth's B-field)
5
0
latitude (deg)
50
100
B-mag
x 10
4
nT
nT
3
0
2
WMM2005
1
WMM2005
Tilted Dipole
Tilted Dipole
-5
-100
-50
0
latitude (deg)
50
0
-100
100
-50
0
latitude (deg)
50
100
Figure 2.4: Comparison of the tilted dipole model and WMM2005 model for a polar
circular LEO orbit at an altitude of 500 km (the longitude is fixed at -71°)
2.3.3. Disturbance Force due to Earth’s Magnetic Field
Another aspect that can be seen from Eq. (2.44) is that the magnitude of the gradient of
the Earth’s magnetic field scales as:
48
∇Be ∼
μ0 μm
Ri4
(2.54)
This results in a value of roughly 10-10 Wb/m3 (for a 500 km orbit) which would result in
a force less than 1 μN on a dipole having a strength of 104 A-m2. Since inter-dipole forces
in an electromagnetic formation are generally of the order of 1 mN, one can safely ignore
the magnetic force on the formation due to Earth’s magnetic field and consider it as a
disturbance force in the control design process. If this force is significant, as compared to
inter-dipole forces, then it can be explicitly taken into account.
2.4. Deep Space 2D Dynamics
In previous sections, the dynamics of electromagnetic formations in LEO was discussed,
which is suitable for deep space missions as well if we drop the gravity terms and neglect
the Earth’s magnetic field. Nevertheless, many deep space missions can be captured
effectively by using simplified 2D dynamics, e.g., in telescope slew maneuvers which are
essentially planar and can be described by 2D dynamics. Moreover, the EMFF testbed
[29] dynamics are also 2D. In these cases, it is much easier to work with the simplified
2D dynamics rather than the full 3D dynamics discussed in last sections. Therefore, in
this section 2D deep space dynamics for an N-satellite electromagnetic formation will be
derived. It is assumed that each satellite in the formation has two identical orthogonal
coils and one reaction wheel to store the angular momentum along the z-direction (see
Figure 2.5).
49
Figure 2.5: EMFF coils, reaction wheel and coordinate frame definition for 2D dynamics
The deep space equations of motion in 2D of the ith satellite in the formation in an
inertial frame are given by (using Eqs. (2.24) and (2.30)):
mi xi = Fxi (x, y, α , μ ) + Fdxi
mi yi = Fyi (x, y, α , μ ) + Fdyi
(2.55)
I iα i = τ im (x, y, α , μ ) − τ iRW + τ di
where mi is the mass of the ith satellite, Ii is the moment of inertia of the satellite about the
z-axis, (xi, yi) define the location of the satellite in the inertial frame (whose center is
assumed to be located at the center of mass of the formation), αi is the rotation angle of
the satellite about the z-axis. The Fd’s and τ d ’s are the disturbance forces and disturbance
torques, respectively, acting on the satellite. Fxi and Fyi are the components of the net
magnetic force while τ im and τ iRW are the net magnetic torque and reaction wheel torque,
respectively. These magnetic forces and torques depend on the location, orientation and
magnetic moments of each satellite in the formation, whose vectors are defined as
follows:
x = [ x1
x2
α = [α1 α 2
xN ]T , y = [ y1
yN ]T
y2
α N ]T , μ = [ μ1x
50
μ1 y
μ Nx
μ Ny ]T
(2.56)
The magnetic forces and torques acting on each satellite in the formation due to
the magnetic moments of the other satellites can be derived by approximating each coil
by a magnetic dipole. This approximation, also known as the far-field approximation,
gives the force and torque, on dipole i due to dipole j, with magnetic moments μi and
μ j , as given by Eq. (2.19) and Eq. (2.35), respectively. The two orthogonal coils give
each satellite the ability to control the direction and magnitude of the magnetic moment
vector in a plane. We define the magnetic moment of a coil as follows:
μic = ni Ai I i
(2.57)
where ni, Ai and Ii are the number of turns, the area enclosed by the coil and the current
flowing through the coil, respectively. For a fixed coil geometry, the magnetic moment is
varied by changing the current, Ii, and thus it acts as the control input for the system. We
can represent the two coils as a single “steer-able” dipole with magnitude (μi) and
direction (βi) as follows:
μi = μix2 + μiy2
β i = tan −1
μiy
μix
(2.58)
This representation helps in writing the force and torque on the ith satellite, due to
the magnetic moment on the jth satellite, more compactly, by expanding Eq. (2.19) and
Eq. (2.35) as follows (refer to Figure 2.6):
Fij = −
3μ0 μi μ j ⎡sin ai sin a j − 2 cos ai cos a j ⎤
⎢
⎥
4π dij 4 ⎣ cos ai sin a j + sin ai cos a j ⎦
(2.59)
μ 0 μi μ j
( cos ai sin a j + 2sin ai cos a j )
4π dij 3
(2.60)
τ ij = −
where Fij is the magnetic force acting on satellite i due to satellite j, τ ij is the torque
acting on satellite i due to satellite j and dij is the distance between the two satellites, i.e.:
51
dij = ( xi − x j ) 2 + ( yi − y j ) 2
(2.61)
The total magnetic force on the ith satellite can be found by summing the force
contributions from all the remaining satellites in the formation, as the principle of
superposition applies to the total magnetic field of the formation. In doing so, it should be
noted that the angles ai and aj in Eqs. (2.59) and (2.60) are measured from a line
connecting the center of masses of the satellites and, the forces Fij are in this rotated
frame of reference. Therefore, we need to transform the forces from this rotated frame to
the global frame by multiplying the force vector with a transformation matrix Mij given
as:
⎡cos θij
M ij = ⎢
⎣ sin θij
− sin θij ⎤
cos θij ⎥⎦
(2.62)
xr
N
aj
yr
γi
N
ai
dipole j
θ ij
dipole i
Figure 2.6: Geometry of steerable dipoles for magnetic force and torque computation.
where:
⎛ y j − yi
⎜ x j − xi
⎝
θij = tan −1 ⎜
52
⎞
⎟⎟
⎠
(2.63)
Substituting the total magnetic force and torque on each satellite into Eqs. (2.55),
we obtain:
⎡sin ai sin a j − 2 cos ai cos a j ⎤ ⎫⎪ ⎡ Fdxi ⎤
⎡ mi xi ⎤ ⎧⎪ N −3μ0 μi μ j
⎢ m y ⎥ = ⎨ ∑ 4π d 4 M ij ⎢ cos a sin a + sin a cos a ⎥ ⎬ + ⎢ F ⎥
i
j
i
j ⎦⎪
⎣ i i ⎦ ⎪⎩ j =1, j ≠i
ij
⎣
⎭ ⎣ dyi ⎦
μ μi μ j
I iα i = ∑ − 0 3 ( cos ai sin a j + 2sin ai cos a j ) − τ iRW + τ di
4π dij
j =1, j ≠ i
N
(2.64)
For an N-satellite formation in 2D there are 3N control inputs where each satellite
has the control vector:
ui = [ μix
μiy τ iRW ]T
(2.65)
Note that in Eqs (2.64) we are using a polar representation of the magnetic moments (Eqs
(2.58)) where the angle β is related to other angles in Figure 2.6 as follows:
γ i = α i + βi = θij + ai
(2.66)
From inspection of Eqs (2.64), it is clear that the magnetic moments appear as
product pairs and hence the right hand side of the equation is a polynomial in the control
inputs with each term having a degree two while the coefficients of these polynomials are
nonlinear functions of the current state of the formation. Another important aspect of
these dynamics is that the magnetic force and torque between two dipoles decays rapidly
with the distance between the dipoles, therefore in close proximity the satellites have
much more control authority as compared to when they are farther apart.
It should be pointed out that since the magnetic forces are only internal to the
system, the linear momentum of the center of mass of the whole formation cannot be
changed. Although this is not a limitation as far as formation flying is concerned (since
for formation flying it is only the relative position and orientation of the satellites that
needs to be controlled), it has a consequence for the dynamics in Eqs (2.64) since two of
53
the equations in this set are actually dependent upon the other equations. This dependence
is also evident from the fact that the sum of the magnetic forces must be zero and hence
one of the satellites has essentially “free” or dependent dynamics. This set of “free”
dynamics needs to be explicitly taken into account while developing any control laws, as
done in the following chapters. This dependence can be explicitly stated as follows:
N
L = ∑ mi v i = constant
i =1
N
H = ∑ ( I iα i + I iRW ωiRW + mi ri × v i ) = constant
(2.67)
i =1
where L is the total linear momentum and H is the total angular momentum of the
formation.
2.5. Relative Translational Dynamics
In this section relative translational dynamics applicable to general EM formations in
LEO will be presented since the formation translational control is based on relative
dynamics which can easily be derived from Eqs. (2.24). Ignoring the J2 terms for
simplicity, the translational dynamics of satellite-i in the ECI frame can be written
compactly using vector notation as:
mi R i +
μe mi Ri
Ri3
m
= F i + F di
(2.68)
Similarly the dynamics of satellite-k are given as:
mk R k +
μe mi Rk
3
k
R
m
= F k + F dk
(2.69)
Subtracting Eq. (2.69) from Eq. (2.68), we obtain the dynamics of satellite-i relative to
satellite-k in the ECI reference frame as:
54
r ki +
μe ( R k + rki )
R k + rki
3
−
μe R k
Rk3
m
m
m
m
F i F k F di F dk
=
−
+
−
= f i − f k + f di − f dk
mi mk
mi
mk
(2.70)
For deriving a relative position control law, it is more advantageous to work in the
non-inertial orbital coordinates FRO as defined previously. For this purpose we will
express the relative dynamics of satellite-i in the orbital reference frame FRO, located at
center of mass of the formation. Although this derivation can also be done in an orbital
frame attached to the “leader” satellite (as done in [32]), the leader satellite will be
maneuvering in general so its orbital velocity would not be a constant; while for
electromagnetic formations, the center of mass of the formation cannot be moved during
maneuvering using Electromagnetic actuation. Therefore, it is more accurate to use an
orbital reference frame attached to the center of mass of the formation as opposed to the
leader satellite.
For simplicity we will assume a circular orbit for the formation for which the
angular velocity of the reference orbit is simply:
FRO
FI
ω = ω0 l z
(2.71)
where ω0 is the orbital mean motion given by Eq. (1.2) (which is constant for a circular
orbit), and lz is a unit vector pointing along the z-axis of frame FRO. Using the Transport
Theorem Eq. (2.1), the inertial acceleration can be transformed to the orbital coordinate
system as:
FI
r=
FRO
r +2
FRO
FI
ω ×r +
FRO
FI
ω ×
FRO
FI
ω ×r
(2.72)
By expressing the relative position of satellite-i in the orbital reference frame FRO as:
rki = xki lx + yki l y + zki lz
55
(2.73)
and using Eqs. (2.70) and (2.71), the relative dynamics of satellite-i with respect to
satellite-k in the orbital reference frame FRO can be written as:
xki − 2ω0 yki − ω02 xki +
yki + 2ω0 xki − ω02 yki +
zki +
where f
m
μe zki
Ri3
μe xki
= f ixm − f kxm + fixd − f kxd
3
i
R
μe ( Rk + yki )
3
i
R
μe
−
Rk2
= fiym − f kym + f iyd − f kyd
(2.74)
= fizm − f kzm + f izd − f kzd
and f d are specific magnetic force components and specific disturbance force
components, respectively, expressed in the orbital frame FRO. These equations will be
used in Chapter 4 for the derivation of translational control laws in LEO.
One difficulty associated with using Eqs. (2.74) is that we need to express all the
terms in the orbital reference frame FRO. Since there is no physical object present at the
location of the center of mass, no direct observations can be made regarding its position
or velocity. Nevertheless, these can be deduced from the combined measurements of the
satellites in the formation as follows.
The location of the center of mass of the whole formation is given as:
N −1
RRO =
∑m R
i =0
N −1
i
i
(2.75)
∑ mi
i =0
Similarly the velocity of the formation center of mass would be given as:
N −1
RRO =
∑m R
i =0
N −1
i
i
∑m
i =0
i
56
(2.76)
We also need to determine the transformation matrix that transforms the vectors
from the ECI frame FI to the orbital frame FRO. Since the y-axis of FRO points in the
direction of the vector RRO , we have:
yˆ RO =
RRO
(2.77)
RRO
where yˆ RO is a unit vector pointing along the y-axis of the frame FRO. Similarly, for a
circular orbit, we have:
xˆ RO = −
RRO
(2.78)
RRO
where xˆRO is a unit vector pointing along the x-axis of the frame FRO. Finally a unit
vector along the z-axis of the frame FRO is given by:
zˆRO = xˆRO × yˆ RO
Let
FI
(2.79)
T FRO be the transformation matrix from the orbital frame to the ECI frame. Then,
we have the following relations:
⎡ vOx ⎤
⎡1 ⎤
⎢ v ⎥ = − FI T FRO ⎢0 ⎥ ,
⎢ Oy ⎥
⎢ ⎥
⎢⎣ vOz ⎥⎦
⎢⎣0 ⎥⎦
⎡ rOx ⎤
⎡0⎤
⎢ r ⎥ = FI T FRO ⎢1 ⎥ ,
⎢ Oy ⎥
⎢ ⎥
⎢⎣ rOz ⎥⎦
⎢⎣ 0⎥⎦
⎡ rOy vOz − rOz vOy ⎤
⎡0⎤
⎢
⎥ FI FRO ⎢ ⎥
⎢ rOz vOx − rOx vOz ⎥ = T
⎢0⎥
⎢ rOx vOy − rOy vOx ⎥
⎢⎣1 ⎥⎦
⎣
⎦
(2.80)
where the parameterization of the unit vectors in the ECI reference frame are defined as:
yˆ RO = ⎡⎣ rOx
xˆ RO = − ⎡⎣vOx
rOy
vOy
T
rOz ⎤⎦ ,
T
vOz ⎤⎦ .
Combining the relations given in Eqs. (2.80) and taking the inverse (or transpose) we
obtain:
57
FRO
T
FI
⎡
−vOx
⎢
rOx
=⎢
⎢ rOy vOz − rOz vOy
⎣
−vOy
rOy
rOz vOx − rOx vOz
⎤
−vOz
⎥
rOz
⎥
rOx vOy − rOy vOx ⎥⎦
(2.81)
Using this transformation matrix we can transform the measurements from the ECI
reference frame to the orbital frame and, the relative dynamics, given by Eqs. (2.74), can
be used to design control laws for controlling the formation in LEO as will be presented
in Chapter 3.
2.6. Summary
This chapter discusses both the translational and attitude dynamics of a general
electromagnetic formation. These dynamics are suitable for simulating a general EM
formation in a circular orbit around the Earth for the purpose of evaluating control laws
that can be derived based on these dynamics. Earth’s magnetic field models are also
discussed that are required for simulating EM formations in LEO. This chapter also
discusses the simplified deep-space 2D dynamics that are suitable for some missions.
Lastly the relative dynamics are derived that are needed for the derivation of the control
laws.
58
Chapter 3
Control of EM Formations in LEO
As discussed in Chapter 1, from a system design point of view, EMFF has the greatest
potential for formations that require high delta-V for formation hold or formation
reconfiguration. High delta-V requirements translate directly into fuel mass for
conventional thruster-based systems and, over long mission lifetimes, EMFF may be the
only option to hold the formation. One such region where EMFF can be superior to using
conventional thrusters is Low Earth Orbit (LEO). In LEO, due to the proximity to Earth,
certain formations require high delta-V over their mission lifetime. Electromagnetic
formations on the other hand can hold the formation in LEO indefinitely, barring any
system failure. Unfortunately one of the complications for EMFF in LEO is the strong
magnetic field of the Earth that induces very-high disturbance torques on the EMFF
satellites and angular momentum management can be an issue. On the other hand the
magnetic field of the Earth can be used as an asset, since the angular momentum
accumulated in the reaction wheels can be dumped into the Earth’s magnetic field [10].
The purpose of this chapter is to formulate a framework in which control laws can
be designed for EMFF in LEO. Within this framework, practical control laws will be
derived for the control of a general electromagnetic formation in LEO. Angular
momentum management will be accomplished by exploiting the nonlinearity of the
magnetic dipoles by defining a “dipole polarity switching control law.” Furthermore,
59
these control laws can be readily extended to deep-space missions where the
complicating factor of Earth’s magnetic field would be absent.
3.1. Control Framework
From models presented in Chapter 2 it is clear that the dynamics of electromagnetic
formations are not only nonlinear, they are coupled as well in the sense that if one of the
dipoles in the formation is changed, every satellite with a non-zero dipole is affected.
Moreover, in LEO Earth’s magnetic field has a strong influence over the attitude control.
Keeping these points in mind, a general framework will be defined here which can serve
as a guideline for designing any form of control for EMFF.
In the design of control for EMFF, the following guidelines must be considered:
•
EMFF can control all the relative degrees of translational motion, i.e. the center
of mass of the formation cannot be moved. Although for formation flying
missions this is not a limitation, this must be considered explicitly in any control
design.
•
The dynamics of EMFF have uncertainty. The dynamic models are based on a
far-field model of the dipoles. These models are accurate only when the distance
between the dipoles is greater than many coil diameters. In the near-field region,
these models overestimate the force and torque exerted on the dipoles. Although,
this is not in itself a limitation, since better models based on the exact Biot-Savart
Law [17] can be built at the expense of computational cost. If it is undesirable to
use exact models then this uncertainty in the dynamic models must be explicitly
taken into account in any control formulation.
60
•
The environment in which electromagnetic formations will operate may be
uncertain. This is especially true in LEO due to the uncertainty present in the
Earth’s magnetic field model (see Section 2.3). Since the disturbance torque due
to Earth’s magnetic field is large, tackling the uncertainty through robustness of
the control laws may be inadequate; in that case direct external measurement of
the Earth’s magnetic field or online adaptation may be used.
•
The dynamics of electromagnetic formations are coupled and the magnetic force
on each satellite is given by a set of polynomial equations, therefore to find the
control inputs dynamic inversion through optimization has to be used as discussed
in Section 4.5.
•
Due to dipole coupling, centralized translational control has to be used.
Therefore, one satellite in the formation may act as the “leader” and solve the
control problem for the whole formation and transmit the translational control
commands to individual satellites in the formation. This may result in a single
point-of-failure in the formation since if the “leader” satellite malfunctions then
the whole formation may be rendered useless. But if the same computational
capacity is built into multiple satellites then this “leadership” role can be handed
over to the other satellites and there would be no single point-of-failure in the
formation.
•
Each satellite is assumed to have its own momentum storage devices, such as
reaction wheels or Control Moment Gyros (CMGs). Therefore, the attitude
control can be implemented in a decentralized fashion and each satellite can
compute its own attitude actuation commands.
61
•
Angular Momentum Management (AMM) can be an issue with using EMFF
especially in LEO. Therefore, any control formulation must take AMM explicitly
into account (e.g. in the dipole equations solution) and verify the angular
momentum storage requirements are met through simulations etc.
3.2. Translational Control for N-Satellite EM Formation
Due to the coupled nature of the dynamics and uncertainty present in the dynamic model,
a centralized adaptive position tracking control scheme will be presented here. Consider a
general N-satellite electromagnetic formation and let the satellites be numbered according
to the index set ℑ = {0,1,..., n} where n = N-1. Designating satellite 0 as the “leader” of
the formation, the translational dynamics of the ith satellite in the formation, relative to
the leader in the orbital frame FRO, are given by Eq. (2.74) and can be written in a
compact vector form as follows:
pi + ci (ω0 , pi ) + g i ( R0 , pi ) = aim + aid
(3.1)
where:
p i = [ x0i
y0 i
z0i ]T = [ pix
piy
piz ]T is the position vector of the ith satellite in
the orbital frame FRO,
ci (ω0 , pi ) = [−2ω0 piy
2ω0 pix
0]T is the coriolis term,
μe pix
⎡
⎤
− ω02 pix
3
⎢
⎥
Ri
⎢
⎥
⎢ μe ( R0 + piy ) μe
⎥
g i ( R0 , pi ) = ⎢
− 2 − ω02 piy ⎥ is the gravity term,
3
Ri
R0
⎢
⎥
⎢
⎥
μe piz
⎢
⎥
3
Ri
⎣
⎦
62
⎡ fixm − f 0mx ⎤
⎢
⎥
aim = ⎢ fiym − f 0my ⎥ is the relative specific magnetic force on satellite-i in FRO and,
⎢ fizm − f 0mz ⎥
⎣
⎦
⎡ f ixd − f 0dx ⎤
⎢
⎥
aid = ⎢ fiyd − f 0dy ⎥ is the relative specific disturbance force on satellite-i in FRO.
⎢ f izd − f 0dz ⎥
⎣
⎦
The relative specific magnetic force on satellite-i, aim , can be defined as a sum of
the far-field approximation and a near-field modifier term as follows:
⎡ fîxm − fˆ0mx
⎢
aim = aˆ im + ⎢ 0
⎢
⎢⎣ 0
⎡ fîxm −
⎢
where aˆ im = ⎢ fîym −
⎢ ˆm
⎢⎣ fiz −
0
fîym − f 0my
0
⎤ ⎡γ ⎤
⎥ ix
0 ⎥ ⎢⎢γ iy ⎥⎥
⎥
fîzm − fˆ0mz ⎥ ⎢⎣γ iz ⎥⎦
⎦
0
(3.2)
fˆ0mx ⎤
⎥
fˆ0my ⎥ is the far-field approximation of the relative specific magnetic
⎥
fˆ0mz ⎥
⎦
force on satellite-i in orbital frame FRO and, γ i = [γ ix
γ iy γ iz ]T is the unknown gain
vector for relative magnetic force. Note that in this formulation of the relative specific
magnetic force, aˆ im is what is known from the model, given by Eqs. (2.13) and (2.19),
and γ i is the unknown multiplication factor that adjusts the forces to the actual level
based on whether the satellite is in the near-field or far-field region. The parameters γ i
represent the uncertainty present in the magnetic dipole strengths and are a function of
the position of the satellites. The rate of change of γ i can be related to the rate of change
of the satellite position using the chain rule as ( γ i )p = ( dγ i dp ) p . By comparing the farfield model force with the exact Biot-Savart law, the term dγ i dp can be estimated for
63
two dipoles in the near-field region. This comparison reveals that these parameters vary
at least an order of magnitude slower than the rate at which the position of the satellites
changes. Moreover, since the desired trajectory is assumed to be sufficiently slowly
varying, the “slowly-varying” assumption for the parameters γ i is a good assumption for
dipoles.
In a similar fashion, treating the unknown relative disturbance forces as slowly
varying parameters di = ⎡⎣ dix
T
diz ⎤⎦ , the relative translational equations for satellite-
diy
i can be written as:
pi + ci (ω0 , pi ) + g i ( R0 , pi ) = aˆ im + Yi θi
(3.3)
Where:
⎡ fîxm − fˆ0mx
⎢
Yi = ⎢ 0
⎢
0
⎣⎢
θi = ⎡⎣γ ix
0
0
fîym − f 0my
0
0
fîzm − fˆ0mz
γ iy γ iz
dix
diy
1 0 0⎤
⎥
0 1 0 ⎥ and,
⎥
0 0 1⎥
⎦
T
diz ⎤⎦ is the parameter vector for satellite-i.
Note that in this formulation, other parameters such as mass etc., if unknown, can also be
incorporated [31], [40].
By defining a relative position vector for the whole formation as
p = [p1 p 2 ... p n ] , the relative dynamics of the whole formation can be written
T
compactly as:
p + C(ω0 , p) + G ( R0 , p) = aˆ m + Yθ
64
(3.4)
where vectors C etc. are obtained by stacking individual vectors ci etc. and Y is a block
diagonal matrix of size 3n x 6n that has Yi on its diagonal.
Consider a trajectory following problem in which each satellite is required to
follow a predefined trajectory that is sufficiently slowly varying (i.e., time constant of the
change in trajectory is much less than that of the system) and sufficiently smooth (i.e.
pd (t ) ∈ C 2 ). Note that the desired trajectory vector p d (t ) = [p d 1 (t ) p d 2 (t )
p dn (t )]T
defines trajectories for the satellites relative to the leader and does not include the
trajectory for the leader itself, which is appropriate for EMFF since the center of mass of
the formation cannot be moved using EMFF. In other words, p d (t ) defines all the
relative translational degrees of freedom of the formation that can be controlled. Let
p(t ) = p(t ) − p d (t ) , p(t ) = p(t ) − p d (t ) and, θ = θˆ − θ be the position error, velocity error
and parameter error vectors, respectively, for the whole formation. Define a formation
composite error vector as follows:
s(t ) = p(t ) + Λp(t )
(3.5)
where Λ > 0 is a diagonal gain matrix. The rate of change of the composite error, s(t ) ,
can be written as:
s(t ) = p(t ) − p r (t )
(3.6)
p r (t ) = p d (t ) + Λp d (t ) − Λp(t )
(3.7)
where:
Consider a Lyapunov function defined as follows [31]:
65
V (t ) = 12 s(t )T s(t ) + 12 θT Γ −1θ
(3.8)
where Γ > 0 is a diagonal gain matrix. The time derivative of the Lyapunov function is
given as:
V (t ) = s(t )T s(t ) + θT Γ −1θ
(3.9)
Using Eqs. (3.4) and (3.6), Eq. (3.9) can be written as:
{
}
V (t ) = sT −C(ω0 , p) − G ( R0 , p) + aˆ m + Yθˆ − p r (t ) − sT Yθ + θT Γ −1θ
(3.10)
By choosing the control law with K p > 0 :
aˆ m (t ) = C(ω0 , p) + G ( R0 , p) − Yθˆ + p r (t ) − K p s(t )
(3.11)
and the adaptation law:
θˆ (t ) = ΓYT s(t )
(3.12)
the rate of change of the Lyapunov function, Eq. (3.10), reduces to:
V (t ) = −sT (t )K p s(t )
(3.13)
which is negative semi-definite and stability, in the sense of Lyapunov, of the whole
formation follows from the Lyapunov theorem. Since, the Lyapunov function and
underlying system are time-varying (due to the time dependent desired trajectory),
LaSalle’s Invariance Principle [30] cannot be used to establish asymptotic stability.
However, asymptotic stability can be established using Barbalat’s Lemma [31] as done in
the following section.
66
3.2.1. Proof of asymptotic convergence using Barbalat’s Lemma
To show that the formation composite error approaches zero asymptotically using the
control law (3.11) and the adaptation law (3.12), Barbalat’s Lemma, which deals with the
asymptotic convergence of functions, will be used and is defined as follows:
Lemma 3.1: Barbalat’s Lemma
If a differentiable function f(t) has a finite limit as t → ∞ , and if its time derivative is
uniformly continuous, then f (t ) → 0 as t → ∞ .
For proof see Reference [31]. Note that a function f(t) will converge to a limit as
t → ∞ if it satisfies the following two conditions:
1.
f (t ) is lower bounded
2.
f (t ) is negative semi-definite
The uniform continuity requirement on a function f(t) can be stated as a
boundedness requirement on its first derivative, or in other words if the function f(t) is
uniformly Lipschitz then its first derivative will be bounded. Note that a function
f:
→
is defined as uniformly Lipschitz if there exists a constant L > 0 such that the
following condition:
f (t1 ) − f (t2 ) ≤ L t1 − t2
for all t1 and t2 in
is satisfied. Using these observations, the following Lemma can be
stated:
Lemma 3.2: Lyapunov like Lemma for Time-Varying Systems
If a scalar function V (t ) satisfies the following conditions:
1.
(3.14)
V (t ) ≥ Vmin , i.e. it is lower bounded,
67
2.
V (t ) ≤ 0 , i.e. it is negative semi-definite,
3.
V (t1 ) − V (t2 ) ≤ L t1 − t2
∀t1 , t2 ∈
for some positive constant L. This condition
is equivalent to the boundedness of V (t ) .
then V (t ) → 0 as t → ∞ .
From Eq. (3.8) it is clear that V (t ) is a positive definite function and hence it is
lower bounded. The second condition of Lemma 3.2 is also satisfied as shown in Eq.
(3.13). To see that the last condition of Lemma 3.2 is satisfied we have:
V (t ) = −2sT (t )K p s(t )
(3.15)
Since V (t ) ≤ 0 and V (t ) is a positive definite function hence Vmin ≤ V (t ) ≤ V (0) which
implies that both s and θ are bounded. Moreover, Eqs. (3.6) and (3.7) and the fact that
the desired trajectory vector p d (t ) is sufficiently smooth ( ∈ C 2 ) implies that s(t ) remains
bounded for any real mechanical system. Given these facts, it can be concluded that V (t )
remains bounded and all the conditions of Lemma 4.2 are satisfied hence V (t ) → 0 as
t → ∞ . From Eq. (3.13) this can only happen when s(t ) → 0 or in other words both p(t )
and p(t ) approach zero as t → ∞ . This completes the proof of asymptotic convergence
of the formation trajectory to the desired trajectory. It should be noted that the above
proof does not imply that the unknown parameter error θ also converges to zero but
rather it only implies that it remains bounded.
68
3.3. Adaptive Attitude Control
In this section an adaptive attitude tracking control law will be presented that can be
derived in a fashion similar to what was done in the previous section for translational
control. Each satellite in the formation is assumed to have three orthogonal reaction
wheels, hence the attitude control can be implemented in a decentralized fashion as
discussed earlier. In the attitude control as well, there are two uncertainties in the model,
namely the actual magnetic torque that acts on the satellite due to the magnetic field of
other satellites is uncertain and can vary as much as 10%, depending upon whether the
satellite is in the near- or far-field region, and the second uncertainty lies with the Earth’s
magnetic field as discussed in Section 2.3.
To account for these model uncertainties, an adaptive control scheme is proposed
with the control and adaptation laws respectively as:
m
e
τ RW = ( H B + H RW ) × ω + τˆ + τˆ + Iωr − Yτ θˆ τ + Kτ sτ
(3.16)
θˆ τ = − Γτ YτT sτ
(3.17)
m
e
where τˆ and τˆ are the estimated torques on the satellite due to other satellite dipoles
and the Earth’s magnetic field, respectively. These can be computed using Eqs. (2.33)
and (2.40). Other symbols are defined as follows: Yτ defined as:
⎡τˆxm 0 0
⎢
Yτ = ⎢ 0 τˆym 0
⎢ 0 0 τˆzm
⎣
μ = ⎡⎣ μ x
μy
0
−μz
μz
−μ y
0
μx
μy ⎤
⎥
− μ x ⎥ is the gain matrix for the parameters θˆ τ ,
0 ⎥⎦
T
μ z ⎤⎦ is the dipole control vector for the satellite,
69
θˆ τ = ⎡⎣γ τ x
γτ y γτ z
Bex
Bey
T
Bez ⎤⎦ ,
Kτ = a positive definite diagonal gain matrix,
sτ (t ) = ω(t ) + Λτ α (t ) is the composite attitude tracking error vector,
ω = ω − ωd ,
α = α − αd ,
Λτ > 0 is a diagonal gain matrix,
ω r = ω d − Λτ α + Λτ α d .
By using a similar procedure as outlined in the previous section for the
translational control, the attitude control law and the adaptation law can be shown to
asymptotically follow a predefined, sufficiently smooth attitude trajectory specified by
α d , ω d and ω d . In this formulation the attitude orientation vector α can be defined by
using Euler angles, in which case the kinematic relationship given by Eq. (2.32) relates
the attitude angles with the body axis angular velocity ω . But for large angle slews, to
avoid singularities present in the Euler angle representation of the attitude, we can use
quaternions, in which case the orientation vector α can be defined using quaternion
rotation axis components, i.e., α = [q1
q2
q3 ]T . In the case of quaternions, error in the
orientation (i.e., α ) needs to be described by two successive quaternion rotations since
the error quaternion will be the current satellite attitude quaternion with respect to the
commanded attitude quaternion [41]. Therefore, the difference operator on the
quaternions used to generate the error vector can be shown to be [22]:
70
⎡ −q1
α = q′ − q′d = ⎢⎢ − q2
⎢⎣ − q3
where
q′ = [ q1
prime
q2
indicates
the
last
q0
q3
− q3
q2
q0
−q1
three
− q2 ⎤
q1 ⎥⎥ q d
q0 ⎥⎦
components
(3.18)
of
the
quaternion
(i.e.,
q3 ] ) and qd is the desired attitude quaternion that can be defined as
T
follows:
⎡ qd 1 ⎤
⎢q ⎥
⎡ cos(ξ / 2) ⎤
⎢
qd = d 2 ⎥ = ⎢
⎢ qd 3 ⎥ ⎣U sin(ξ / 2) ⎥⎦
⎢ ⎥
⎣ qd 4 ⎦
(3.19)
where U is a unit vector about which the angle ξ defines a positive rotation. In the
quaternion formulation, the kinematic relation given by Eq. (2.31) (by using the error
quaternion in place of q) can be used to compute ωr as required in the control law given
by Eq. (3.16).
3.4. Angular Momentum Management
As discussed earlier, angular momentum buildup in each satellite in the formation can be
an issue with EMFF. Since we have a “free-dipole” available, we can use it to solve the
dipole equations in such a way as to minimize this buildup of angular momentum in each
satellite as described in Section 4.5. But for near Earth operations, where a constant
disturbance torque may act on each satellite due to the Earth’s magnetic field, the
reaction wheels on each satellite can quickly become saturated even after adjusting the
dipole solution for minimizing angular momentum buildup. In this section, two methods
will be discussed for managing the angular momentum. The first method is applicable for
71
a two satellite formation only, while the second method is general and can be applied to
arbitrarily sized formations.
3.4.1. AMM for two-Satellite Formation using Sinusoidal Excitation
One way of managing the angular momentum for two satellites is the method first
proposed in Reference [11]. In this method the dipoles on each satellite are excited by a
fixed sinusoidal current source that has a frequency much higher than the orbital
frequency of the formation. The desired force between the dipoles is adjusted by varying
the phase difference of the sinusoids on the two dipoles. To illustrate the effect, consider
the force between two sinusoidally excited dipoles in the rotated frame as given by (using
Eq. (2.20)):
Fr
⎡ 2 μ1x μ2 x − μ1 y μ2 y − μ1z μ2 z ⎤
3μ0 sin(ωt ) sin(ωt + φ ) ⎢
⎥
F12 (r12 , μ1 , μ2 ) =
− μ1x μ2 y − μ1 y μ2 x
⎢
⎥
4π r124
⎢⎣
⎥⎦
− μ1x μ2 z − μ1z μ2 x
(3.20)
where ω is the excitation frequency and φ is the phase difference between the two
dipoles. The average force on dipole-1, due to the magnetic field of dipole-2, can be
computed by taking the integral of Eq. (3.20) over one time period (during which we
assume that the individual dipole components remain constant):
Fr
where
Fr
⎡ 2 μ1x μ2 x − μ1 y μ 2 y − μ1z μ 2 z ⎤
3μ0 cos(φ )T ⎢
⎥
F (r12 , μ1 , μ2 ) =
− μ1x μ 2 y − μ1 y μ 2 x
4
⎢
⎥
8π r12
⎢⎣
⎥⎦
− μ1x μ 2 z − μ1z μ 2 x
A
12
(3.21)
F12A is the average force on dipole-1 due to dipole-2 in the rotated frame and T
is the time period of the sinusoidal excitation. The above equation clearly shows that by
adjusting the phase difference between the dipoles, the magnitude of the force can be
72
changed as desired while the orientation can be adjusted by varying the individual
components of the dipole (i.e. by adjusting the currents on individual coils).
The advantage of using this scheme of exciting the dipoles is that the disturbance
torque on each dipole due to the Earth’s magnetic field averages out as can be seen from
Eq. (2.34):
A
T
τ 1E = ∫ μ1 (t ) × BE ( R1 )dt = 0
0
(3.22)
where it is assumed that the Earth’s magnetic field is constant over a time period of the
dipole excitation which will be a good assumption if this time period is much smaller
than the orbital period. The limitation of this method is that it is not clear how it can be
applied to more than two-satellite formations.
3.4.2. AMM using Dipole Polarity Switching
To manage angular momentum for general electromagnetic formations, the unique
nonlinearity of magnetic dipoles can be utilized, namely the force acting on a dipole due
to another dipole depends upon the product of the individual dipole values. Therefore, if
both dipoles change their sign (i.e., flip by 180º) simultaneously and instantaneously, the
force acting on the dipoles does not change. But the torque acting on the satellite due to
Earth’s magnetic field changes sign, as can be seen from Eq. (2.40), resulting in a net
cancellation of the effect of the Earth’s magnetic field on average.
The angular momentum storage capacity required is approximately inversely
proportional to the switching frequency. To see this, the Earth’s magnetic torque is
related to the reaction wheel angular momentum as follows:
E
dH RW
=τ E
dt
73
(3.23)
where:
τ E = Magnetic torque on the satellite due to Earth’s magnetic field
E
H RW
= Reaction Wheel angular momentum due to Earth’s magnetic field
Using Eq. (2.34) and integrating Eq. (3.23) over half of the switching period T, the
reaction wheel angular momentum accumulated over half the switching period is given
as:
E
H RW
=∫
T /2
0
μ (t ) × BE ( R1 )dt
(3.24)
For simplicity, assuming that the dipole strength and Earth’s B-field remain constant
during the switching period, the RW momentum will be given by:
{
} T2
E
= μ (t ) × BE ( R1 )
H RW
(3.25)
Since RW angular momentum is proportional to the size of the RW, we can write:
RW Size ∝
1
Switching Frequency
(3.26)
Other parameters that affect the RW size are formation orbit, formation geometry and
nominal values of the dipoles required to hold the formation.
Practically speaking this “switching” action will take finite time, depending upon
the size of the coils etc., but still it can be accomplished in a relatively small time as
compared to the time constants of the formation error rates. This can be realized by using
capacitors and active switching techniques. The effectiveness of this technique in
managing angular momentum is shown by the simulation results in the next section.
74
3.5. Simulation Results
In order to verify the validity of the control laws and the angular momentum management
method developed in the last sections, a full 6-DOF (Degree Of Freedom) nonlinear
simulation was built using Simulink and Matlab. Using these simulation tools, general Nsatellite electromagnetic formations can be simulated around Earth in the ECI reference
frame including J2 perturbations. A full 6-axis controller enables the control of general
formations that allows for prescription of trajectories with respect to the leader of the
formation (satellite 0). This simulation also includes a tilted-dipole model of the Earth’s
magnetic field. The purpose of these simulation tools is to verify the control concepts
developed in this chapter. For the testing of the EMFF controllers in a more realistic
simulation environment, parallel work is underway to integrate these controllers into the
NASA Goddard FFTB (Formation Flight Test Bed) [76]. See Appendix A for a more
detailed discussion of the interfaces of these simulation tools. Using these simulation
tools, simulation results will be presented to show the viability of using EMFF for LEO
missions.
3.5.1. Two-Satellite Cross Track Formation
In this section simulation results will be presented for a two-satellite formation in a
circular polar orbit around Earth at an altitude of 500 km (see Figure 3.1). The two
satellites are to be kept at a fixed distance from each other in cross-track position while
one face of each satellite needs to point towards Earth at all times.
75
Figure 3.1: Two-satellite formation orbits.
It should be noted that the orbits of the individual satellites are non-Keplerian and
they need to be kept on their respective orbits by continuously applying the necessary
force along the cross-track axis. Such a formation can be used in a SAR (Synthetic
Aperture Radar) application, where the cross-track distance between the two satellites can
act as the baseline of the interferometer and can yield information about ground elevation
differences [5]. Different parameters used in the simulations are listed in Table 3.3-1.
Moreover, a constant unknown external magnetic field component, having a strength
equal to [1x10-5 -0.5x10-5 -2.0x10-5]T Wb/m2 was assumed in the simulation. Similarly a
time varying relative disturbance force of 1mN was also used in the simulation.
76
Table 3.3-1: Parameter values used in the simulation
Parameter
Cross-track distance (m)
Value
10
Parameter
Value
Kτ
100
Λτ
1
Leader Mean Motion ω0
(rad/s)
Mass of each satellite (kg)
1.1068x10-3
250
Γ fm (gain for γ i 's )
1
I (kg-m2, same for all axes)
20
Γ fd (gain for di’s)
0.1
Kp
0.5
Γτ m (gain for τ x 's )
1e-3
Λ
0.02
Γτ d (gain for Be 's )
1e-7
1× 10−3 sin(ω0 t )
Unknown Earth’s B x1.0x10-5
Relative Dist. force
component (T)
Fdx (N)
-5
0.0
Unknown Earth’s B y-0.5x10
Relative Dist. force
component (T)
Fdy (N)
Unknown Earth’s B z-2.0x10-5
Relative Dist. force Fdz 0.0
component (T)
(N)
In this simulation a switching time of 50 sec. was used for angular momentum
management. Figure 3.2 gives the position error, attitude, control currents, and reaction
wheel angular momentum of the follower satellite for one orbital period. Since the
satellite needs to point towards Earth at all times, one angle (the2 in Figure 3.2) changes
linearly for a circular orbit. Figure 3.3 gives the corresponding quaternion values for the
follower satellite. Figure 3.3 also gives the algebraic difference between the actual and
desired quaternions of the follower satellite (note that it is not the attitude error
quaternion). From the graphs of the control currents of the follower satellite the switching
action is evident, while from the graphs of the reaction wheel angular momentum it is
clear that the angular momentum buildup was limited by the switching action. Otherwise
angular momentum would buildup continuously saturating the reaction wheels quickly.
77
1
q2, 100*q2e
q1, 100*q1e
Figure 3.2: Follower satellite results.
0
-1
0
0.5
1
1.5
1
0
-1
0
0.5
1
0
-1
0
0.5
1
1.5
1
1.5
time (h)
q4, 100*q4e
q3, 100*q3e
time (h)
1
1.5
1
0
-1
0
0.5
time (h)
time (h)
Figure 3.3: Follower satellite quaternions.
78
Figure 3.4 gives the values of different estimation parameters. The near-field
adjustment parameters are not given since, for this simulation, the satellites are in the farfield region and these parameters are zero. As can be seen from these figures, the
adaptation laws effectively vary the estimated values of the parameters in order to adjust
for unknown external disturbance forces and torques.
Figure 3.4: Different estimated parameters
The results given above demonstrate the viability of the control algorithm and
also the effectiveness of the dipole polarity switching control law as the angular
momentum is managed efficiently.
3.6. Summary
In this chapter a control framework has been presented, which can be used to design
control laws for general EM formations for specific missions. Using this framework, 6-
79
DOF trajectory following adaptive control laws have been derived that are suitable for
controlling general EM formations in LEO. These control laws specifically account for
the uncertainty present in the dipole model and the Earth’s magnetic field model. The
angular momentum management for a two-satellite formation using phase control was
introduced. For general EM formations, the angular momentum management using dipole
polarity switching was presented. Proof of concept of these control laws was presented
using a fully nonlinear simulation for a circular polar orbit in LEO. This chapter presents
practical algorithms, with low computational burden, that can be used to track trajectories
for general EM formations in LEO. In Chapter 5, algorithms for generating optimal
trajectories will be presented that can be tracked in real-time by using the control laws
presented in this chapter.
80
Chapter 4
APFM and Stability Analysis
In this chapter the Artificial Potential Function Method (APFM) will be used for
generating collision-free trajectories for general EM formations. This method has a low
computational burden as compared to optimal trajectory generation algorithms. Although
it was asserted in Chapter 1 that a fully actuated EM formation can control all the relative
degrees of freedom, it is important to present a practical algorithm that can be used to
reconfigure an EM formation from a given initial state to a given terminal state. By
combining APFM with Lyapunov theory, an algorithm is presented in this chapter that
can be used to reconfigure general EM formations asymptotically while avoiding
collisions. At the end of the chapter methods of solving the dipole equations will also be
presented.
4.1. Artificial Potential Function Method for Vehicle Guidance and
Control
The Artificial Potential Function Method (APFM) is an algorithm that allows the
guidance and control of autonomous vehicles with very little computational burden and
hence is suitable for real-time control of complex systems such as EMFF. This method is
most popular for the terrestrial robotic trajectory planning and is based on the design of a
potential function with a global minimum at the desired configuration of the system and
81
maxima at the location of obstacles [33], [34]. This potential function is used to formulate
a Control Lyapunov Function to generate a feedback control law that can reconfigure the
system from a given initial state to a desired terminal state while avoiding collisions. The
usefulness of this method stems from the fact that it combines the functions of guidance
and control, with a low computational burden, and generates a feedback control law with
robustness guarantees. This method will be developed in this section for a 2D, twovehicle electromagnetic formation for the purpose of illustration and application to
EMFF.
4.1.1. An Illustrative Example
Consider a two-vehicle electromagnetic formation on a flat floor and let one of the
vehicles be labelled 0 while the second one is labeled 1. The relative translational
dynamics of vehicle 1 with respect to vehicle 0, in the global inertial frame, are given as
(using Eq. (2.55)):
mrx1 = 2 Fx1
mry1 = 2 Fy1
(4.1)
where:
rx1 = x1 − x0 is the relative x-position of vehicle 1 w.r.t. vehicle 0
ry1 = y1 − y0 is the relative y-position of vehicle 1 w.r.t. vehicle 0
F1m = ⎡⎣ Fx1
T
Fy1 ⎤⎦ is the magnetic force acting on vehicle 1
m = mass of vehicles 0 and 1 (assumed identical).
In writing Eq. (4.1) the following fact is used:
F0m + F1m = 0
82
(4.2)
since linear momentum of the formation is not changed. Note that in this formulation the
translational dynamics of vehicle 0 are completely dependent on those of vehicle 1, i.e.
vehicle 0 moves accordingly such that the center of mass of the system remains fixed in
the absence of external perturbations.
Consider a reconfiguration problem in which the formation needs to be moved
from a given initial state to a known terminal state. It is also desirable that there be no
collision between the vehicles. To achieve these objectives using APFM, a potential
function is defined as follows:
φ (t ) = 12 r1T Mr1 + O(r1 )
(4.3)
where:
r1 = relative position vector of vehicle 1 = [rx1
T
ry1 ] ,
r1 = r1 − r0 is the position error for vehicle 1,
r0 = desired terminal location (a constant),
M = a constant symmetric positive definite scaling matrix,
O(r1 ) is an obstruction function that is defined such that it has a large value in the
avoidance region around the obstruction and is zero outside.
The obstruction function can be constructed in many ways, e.g. as an exponential
and a simple distance function:
(
O(r1 ) = λ1 exp −λ2r1T Nr1
O(r1 ) = c r1
−1
)
(4.4)
where λ1 , λ2 and c are positive shaping constants for the obstruction function and N is a
symmetric positive-definite shaping matrix. The obstruction function is added to prevent
any collision between the two vehicles by creating a “repulsive force” whenever the two
83
vehicles come near each other. A typical potential function, using an exponential, is
shown in Figure 4.1.
Figure 4.1: A potential function shape for vehicle 1. The obstruction function is at the
location of the vehicle 0.
The potential function defined in Eq. (4.3) is used to construct a Lyapunov
Function [31] as follows:
V (r1 , r1 ) = 12 r1T Pr1 + φ (r1 )
(4.5)
The time derivative of Eq. (4.5) defines the rate of descent at a given point in the statespace and is given as:
V (r1 , r1 ) = r1T Pr1 + r1T Mr1 + ∇O (r1 )r1
(4.6)
If we choose the acceleration r1 (with K > 0 ) as:
r1 = − P −1 ( K (r1 , r1 )r1 + Mr1 + ∇OT )
then Eq. (4.6) becomes:
84
(4.7)
V (r1 , r1 ) = −r1T K (r1 , r1 )r1 ≤ 0
(4.8)
which is negative semi-definite and, according to the Lyapunov Theorem [30], the system
is stable.
To prove asymptotic stability, LaSalle’s Theorem or Invariance Principle [30][31] has to be used (see Section 4.3 for detail). By combining Eq. (4.7) with the dynamics
given by Eq. (4.1), the following control law can be obtained:
F1m (r1 , μ 0 , μ1 ) = − 12 mP −1 ( K (r1 , r1 )r1 + Mr1 − ∇V T )
(4.9)
This control law is a set of two coupled nonlinear polynomial equations in the unknown
control inputs μ 0 = ⎡⎣ μ0 x
T
μ0 y ⎤⎦ and μ1 = ⎡⎣ μ1x
T
μ1 y ⎤⎦ . For ways of solving this equation
see Section 4.5 later in this chapter.
The simulation results for a two-vehicle formation using this method are shown in
Figure 4.2. In this simulation masses for both the vehicles were assumed to be 30 kg and
the control current was saturated at 100 A. Each vehicle was assumed to have two
orthogonal coils with nA = 100 A-turns. The controller gains used were assumed to be
diagonal and a value of 1 for K and 0.01 for M were used. Figure 4.2 gives the time
histories of the state and control variables for both the vehicles in the formation with
label-0 for vehicle-0 and label-1 for vehicle-1.
Figure 4.3 shows the values of the potential function along the trajectory of
vehicle-1 and clearly shows the descending nature of the potential function (phi) as the
vehicle reaches its desired position.
85
Figure 4.2: Reconfiguration results for a two-vehicle formation using APFM.
Figure 4.3: Values of the potential function along the trajectory of vehicle-1.
86
4.1.2. Feedback Motion Planning using APFM and its Limitations
From the above illustrative example it is clear that the APFM generates a complete
solution to the motion planning problem. In this section some fundamental issues related
to the motion planning using APFM will be discussed. Generally the motion planning
algorithm can be divided into three distinct sections as shown in Figure 4.4 [35]. Given
the geometry of the satellite formation, obstacles, initial state, and terminal state a
collision free geometric curve is found in the “path-planning” part of the algorithm that
completely ignores the dynamics of the formation. Secondly, given this collision free
path, a reference trajectory is determined for the formation that is based on the time
parameterization of the collision free path. This reference trajectory is such that the
“dynamics” of the formation can actually execute it using trajectory following
algorithms, which is the last piece of this algorithm.
Path Planner
Trajectory Generator
Trajectory Follower
Figure 4.4: The decomposition of the motion planning problem.
The feedback motion planning algorithm, on the other hand, combines all these
functions and generates a feedback plan which produces control inputs for the satellites in
the formation for each state in the state space such that the formation reaches its desired
or goal states. The Artificial Potential Function Method (APFM) generates the feedback
plan by defining a potential function that has simultaneously the properties of a
navigation function [36] and a Lyapunov Function. The gradient of such a function
defines a vector field that defines the descent directions towards the global minimum as
given by Eqs. (4.6) and (4.7). Since we have maxima defined at the locations of
87
obstacles, the vehicles or satellites avoid these obstacles automatically as they move
along these descent directions.
This formulation has at least three limitations. One well documented limitation is
the existence of local minima in the potential function. For example the very simple
potential function defined in Eq. (4.3) has a gradient given by:
∇φ = Mr1 + ∇O(r1 )
(4.10)
Extrema exist for the potential function at the stationary points where the gradient is zero
or in other words at the points in state space where the following equation is satisfied:
Mr1 = Mr0 − ∇O(r1 )
(4.11)
One obvious solution of the above equation is the global minimum r0 but
unfortunately, there is a thin set in state space where the descent gradient Mr1 due to
position error may equal the obstruction function repulsion gradient or ∇O(r1 ) . This thin
set forms a set of saddle points in the state space where there is a possibility that the
satellite will get stuck. To visualize this set in 2D, for a two-satellite formation, see
Figure 4.5.
Global minimum
Contour of obstruction
function
Location of sat. 2
gmt
Gradient of descent
term
Gradient of
obstruction function
Location of sat. 1
Figure 4.5: Case when the gradient of potential function can be zero.
88
It is clear that the gradient of the potential function can be zero only for the case when the
global minimum and the obstruction function are aligned with the descent gradient vector
of the satellite.
One way to get around the local minima is to add an obstruction maneuvering
term or circulation to the potential function gradient whenever the satellite is approaching
such a minima. The local minimum will be approached by the satellite whenever the
gradient of the potential function approaches zero:
Mr1 + ∇O(r1 ) 2 ≤ ε
(4.12)
where ε is a small positive number and . 2 represents the 2-norm. Whenever the
condition given by Eq. (4.12) is satisfied, an obstruction maneuvering term can be added
to the potential function gradient to maneuver around the obstruction. This term is added
such that it is orthogonal to the current velocity vector of the satellite (which would be
parallel to the descent direction given by ∇φ (r1 ) unless an obstruction is present):
g mt ir1 = 0
(4.13)
where g mt is the maneuvering term. This method was used in the illustrative example
given above. The example was constructed such that the satellites would get stuck at the
local minimum as can be seen from Figure 4.2 (since they approach each other head on).
One way of preventing local minima in the potential function is to construct the
potential function in such a way that it has only one unique global minimum. See
Reference [35] for one such method. Another method is to generate the potential function
as a harmonic function which is the solution of Laplace’s equation [37]. It should be
noted that all these methods require a-priori computation of the potential function over
89
the geometry of the problem which may become computationally expensive for complex
problems.
The second limitation of APFM is that it can result in the saturation of the
actuators. Limited control authority implies that the desired acceleration at each point in
the state-space must remain less than or equal to what the system can actually achieve.
This can be ensured by appropriately shaping the potential function in the regions where
the acceleration requirements might be higher due to larger position errors. However,
doing that for a nonlinear actuation system such as EMFF is difficult and, for simplicity
in the above example, a simple saturation was used, which with sufficient damping, gives
reasonable results in the simulations.
The third and most important of the limitations of the APFM method is that it
generates non-optimal trajectories. For space missions, the optimality of the trajectories is
very important since non-optimality directly translates into penalties to the mass metric
for the propulsion system. Specifically for EMFF, since the cost of using control is zero
for a designed system, the trajectories for EMFF are time optimal trajectories. Using
APFM results in trajectories which are sub-optimal hence, for EMFF this method is not
suitable from a system design point of view (see Section 6.4 for a comparison of APFM
with the optimal trajectory results).
Nevertheless, from the point of view of the computational complexity, APFM is
much more efficient as compared to optimal trajectory generation and implementation.
Moreover, APFM can generate feasible trajectories that can be optimized by nonlinear
optimization techniques as discussed in Chapter 5. Thus feasible trajectories, for quite
complex formation maneuvers, can be generated by APFM and used as initial starting
90
guesses for the nonlinear optimizer. Using the potential function shaping method, as
discussed in Section 4.4, trajectories with reasonable convergence times can be generated
with very low computational burden.
4.2. Stability Analysis of EM Formations
In this section, a brief stability analysis of EM formations will be presented for the
purpose of showing that linear stability analysis is inadequate for general EM formations.
In the next section, nonlinear tools based on Lyapunov theory and APFM will be used to
generate an algorithm that can be used to reconfigure a general EM formation.
4.2.1. Limitations of Linear Analysis of EMFF
As discussed in Chapter 1, previous work using linearization techniques has established
stability results for two-satellite formations [9], [28]. Nevertheless, linearization
techniques are inadequate to establish stability results for arbitrary formations. This can
be seen from an inspection of Eqs. (2.64), which shows that the control inputs (i.e. μ 's )
enter the dynamics as product pairs. Hence during linearization, unless these control
inputs have some nominal values, the corresponding entry of the resulting B-matrix
would be zero, which can make the linearized dynamics uncontrollable. To see that this is
indeed the case, consider a two-satellite formation in 2D (the case of the EMFF testbed
[29]). For simplicity, assume that one of the vehicles is fixed as shown in Figure 4.6.
91
y
α
B
A
x
Fixed sat. at (0,0)
Moving sat. at (x0,y0) nominally
Figure 4.6: Coordinate axes for testbed dynamics.
The translational dynamics of the moving vehicle are given as:
⎡ Fabx ⎤
⎢
⎥
⎡ xa ⎤ ⎢ ma ⎥
⎢ ⎥=
⎣ ya ⎦ ⎢ Faby ⎥
⎢
⎥
⎣ ma ⎦
(4.14)
where ma is the mass of the moving satellite. Using Eqs. (2.64), the above equations can
be expanded as follows:
⎡ xa ⎤
3μ0
⎢y ⎥ =
7
⎣ a ⎦ 4π ma ( xa2 + ya2 ) 2
where μa = ⎡⎣ μax
⎡ − μbx ( (2 xa3 − 3xa ya2 ) μax + (4 xa2 ya − ya3 ) μay ) ⎤
⎢
⎥
⎢ μbx ( (−4 xa2 ya + ya3 ) μax + ( xa3 − 4 xa ya2 ) μay ) ⎥
⎣
⎦
(4.15)
T
μay ⎤⎦ is the dipole vector (control input) for the moving satellite while
μb = [ μbx 0] is the dipole vector for the fixed satellite (for simplicity it is assumed that
T
the fixed satellite has only one coil activated with a fixed current). It is also assumed for
simplicity that the angle of the satellites is fixed at zero. Linearizing Eqs. (4.15) about the
nominal point, one can obtain:
92
⎡ xa ⎤ ⎡ 0
⎢ x ⎥ ⎢0
⎢ a⎥ =⎢
⎢ ya ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣ ya ⎦ ⎣ 0
1 0
0 0
0 0
0 0
0
⎡
⎢
2
2
0 ⎤ ⎡ xa ⎤
⎢ x0 (−2 x0 + 3 y0 )
0 ⎥⎥ ⎢⎢ xa ⎥⎥ 3μ0 μbx0 ⎢ ( x02 + y02 )7 / 2
+
⎢
1 ⎥ ⎢ ya ⎥ 4π ma ⎢
0
⎥⎢ ⎥
⎢ y (−4 x 2 + y 2 )
0 ⎦ ⎣ ya ⎦
⎢ 0 2 0 2 7 / 20
⎢⎣ ( x0 + y0 )
⎤
⎥
y0 (−4 x + y ) ⎥
⎥ ⎡ Δμax ⎤
( x02 + y )
⎥⎢
⎥
0
⎥ ⎣ Δμay ⎦
x0 ( x02 − 4 y02 ) ⎥
⎥
( x02 + y02 )7 / 2 ⎥⎦
0
2
0
2
0
2 7/2
0
(4.16)
where μbx 0 is the nominal value of the dipole on the fixed vehicle, while the nominal
value of the dipole on the moving vehicle is assumed to be zero. Clearly, if μbx 0 is zero
then the B-matrix in Eqs. (4.16) would become zero making the system uncontrollable.
Although controllability of the system can be proven for non-zero μbx 0 , there are
situations in which a constant nominal value of the dipole is undesirable. For example in
LEO, a constant value of dipole may result in a constant disturbance torque acting on the
satellite due to the Earth’s magnetic field (see Chapter 2 Section 2.3), which is highly
undesirable. Therefore, in such situations we need control schemes that do not require a
constant nominal value of the dipole. Although, from linear stability analysis the system
appears to be uncontrollable for zero nominal values of the dipole, it is not the case as
revealed by the full nonlinear analysis in the following sections.
As highlighted by the above example, the linear stability analysis tools are
insufficient for EMFF. Some formation configurations, such as TPF (Terrestrial Planet
Finder), would need a constant nominal value of controls to keep the array in a spinning
formation and linear tools can be used. Nevertheless, in general that may not be the case,
for example with the rest-to-rest reconfiguration of a formation in LEO. Therefore, to
establish results for general formations, nonlinear stability analysis tools need to be used.
93
4.3. Reconfiguration of EM Formations using APFM
The purpose of this section is to present an algorithm for the reconfiguration of a fullyactuated EM formation. In the reconfiguration problem, the formation needs to be
reconfigured from a known initial state to a known terminal state with the following
assumptions:
1.
Well-posedness,
2.
Full actuation,
3.
Convex satellite shapes,
4.
Perfect knowledge of full state and dynamics.
Well-posedness assumption is required since EM actuation cannot change the
total linear momentum of the formation. Therefore, any formation reconfiguration
problem that requires a net change in the formation linear momentum does not belong to
well-posed problems for EMFF. This assumption also excludes formations that require
satellites on top of each other and other such scenarios in both the initial and terminal
conditions. Full actuation means that, in
3
, each satellite has three orthogonal coils and
three orthogonal reaction wheels, therefore enabling the control of all the relative degrees
of freedom. Convex shape for the satellites is required since the obstruction functions
used in the algorithm are elliposiodal. Although, the satellite itself can be of any shape,
the reconfiguration problem assumes that the collision avoidance region is an ellipsoid
centered at each satellite in the formation. Lastly perfect knowledge of state and
dynamics is assumed, nevertheless the algorithm is based on Lyapunov function which is
inherently robust and the method will work for small uncertainties.
94
Let the satellites in the formation be numbered from 0 to N-1. We can write the
translational dynamics of the N-satellite EM formation, with respect to the formation
center of mass, in an inertial frame as follows (see Eqs. (2.24)):
pi = vi
v i = fi (p, v i , μ)
where fi :
3N
×
3
×
3N
→
3N
(4.17)
i=0…N-1
represents the nonlinear dynamics of the ith satellite in
the formation, pi is the position vector of satellite-i, v i is the velocity vector of satellitei, p is the position vector of the whole formation and μ is the control vector of the whole
formation.
Define a scalar potential function for satellite-i with a unique global minimum (at
the desired final configuration) as follows:
φi (p) = 12 pi T M i pi + 12
N −1
∑ ( λ exp{−λ (p
j = 0, j ≠ i
1
2
i
)
- p j )T N(p i - p j )}
(4.18)
where:
pi = pi − pi 0 is the position error for the ith satellite
pi 0 = desired terminal location
Mi = a constant symmetric positive definite scaling matrix,
λ1 , λ2 = positive scaling constants for the obstruction function
N = a constant symmetric positive definite matrix for the obstruction function
Such a potential function for a four-satellite formation (in 2D) is shown in Figure
4.7.
95
Figure 4.7: A potential function for one satellite in a 4-satellite formation.
Using these N potential functions (one for each satellite in the formation), the
formation Lyapunov Function can be written as follows:
N −1
⎡
V ( v, p) = ∑ ⎢ 12 v i T Pi v i + 12 p i T M i p i + 12
i =1
⎣
N −1
∑ ( λ exp{−λ (p
1
2
i
j = 0, j ≠ i
)
⎤
- p j )T N (p i - p j )} ⎥
⎦
(4.19)
where the Pi’s are symmetric, positive-definite scaling matrices. Note that the Lyapunov
function does not include a potential function, of the form given by Eq. (4.18), for
satellite-0. This is appropriate for EMFF since total linear momentum of the formation
cannot be changed using EM actuation as discussed earlier.
The time derivative of Eq. (4.19) can be written as:
N −1 ⎡
N −1
⎤
V ( v, p) = ∑ ⎢ v i T Pi v i + p i T M i v i + ∑ ∇Oij (pi , p j ) ( v i − v j ) ⎥
i =1 ⎣
j = 0, j ≠ i
⎦
(4.20)
where:
{
}
∇Oij (pi , p j ) = −λ1λ2 exp −λ2 (pi - p j )T N(pi - p j ) (pi - p j )T N
96
i≠ j
(4.21)
is the gradient of the obstruction function of satellite-i due to satellite-j (note that in the
above equation the gradient is defined as a row vector). From Eq. (4.21) it is clear that the
gradient of the obstruction function is a skew-symmetric function, i.e.:
∇Oij (pi , p j ) = −∇O ji (p j , pi )
i≠ j
(4.22)
Using the above relationship, Eq. (4.20) can be written as:
N −1 ⎡
N −1
⎤
V ( v, p) = ∑ ⎢ v i T Pi v i + p i T M i v i + 2 ∑ ∇Oij v i − ∇Oi 0 ( v i + v 0 ) ⎥
i =1 ⎣
j = 0, j ≠ i
⎦
(4.23)
As discussed in Section 4.1.2, the potential function defined by Eq. (4.18) can
possibly have a thin set in the state-space where the satellites can get “stuck” in the local
minima. Define a set Ti for each satellite (except satellite-0) comprising of the
neighborhoods of all such local minima as follows:
⎧
⎫
N −1
mi
⎪
⎪
T
T
Ti = ⎨ Bε (pi ) M i p i + 2∑ ∇Oij − ∇Oi 0 +
κ = 0, pi ≠ p i 0 ⎬
m0
j =0
⎪
⎪
j ≠i
⎩
⎭
(
)
where Bε (pi ) defines a sphere of radius 0 < ε
1 centered at p i , κ ∈
(4.24)
3
is defined
below, and mi is mass of the ith satellite. In order to avoid the local minima, a hybrid
switching * control law for each satellite (except satellite-0) is defined as follows:
⎧
⎫
N −1
mi ⎪
⎪
T
T
v i = −Pi ⎨K i (pi , v i ) v i + M i pi + 2∑ ( ∇Oij ) − ∇Oi 0 +
κ ⎬ if pi ∉ Ti
m0 ⎪
j =0
⎪
j ≠i
⎩
⎭
(4.25)
⎧
⎫
N −1
mi
⎪
⎪
T
T
v i = −Pi ⎨K i (pi , v i ) v i + M i pi + 2∑ ( ∇Oij ) − ∇Oi 0 +
κ + γ i g imt ⎬ if pi ∈ Ti
m0
j =0
⎪
⎪
j ≠i
⎩
⎭
(4.26)
−1
−1
*
See References [79] and [80] for an introduction to hybrid and switched system stability theory.
97
where Ki is a positive-definite control gain scaling matrix that may be dependent on the
current state to tailor the descent rate according to the position and velocity of the
satellite, γ i is a positive scaling constant, and g imt ∈
3
is a unit vector orthogonal to the
velocity of satellite-i:
g imt i v i = 0,
g imt = 1
(4.27)
The switching control law is defined such that if the satellite-i is not in the neighborhood
of the local minima, i.e. not in the set Ti , then control is computed according to Eq.
(4.25), and if the satellite-i is in the set Ti then Eq. (4.26) is used. Note that if velocity of
satellite-i is zero then the term g imt is not well-defined. This is the reason that the
formations with their initial or terminal conditions at or near to the local minima of the
potential function are not considered as well-posed for this algorithm.
With this selection of the accelerations, the time derivative of the Lyapunov
function can be written as:
N −1
N −1
1
V ( v, p) = −∑ v K i v i − ∑ ∇Oi 0 v 0 −
m0
i =1
i =1
N −1
T
i
N −1
V ( v, p) = −∑ v iT K i v i − ∑ ∇Oi 0 v 0 −
i =1
i =1
1
m0
N −1
∑m v κ
i =1
if pi ∉ Ti , i = 1...N − 1
T
i
i
N −1
∑ m v κ + ∑γ
i =1
i
T
i
k∈I
k
v Tk g kmt ∀ p k ∈ Tk
(4.28)
(4.29)
where the index set I is defined such that it contains the indices of all the satellites for
which the control law given by Eq. (4.26) is used. By construction, the obstruction
maneuvering term g imt is always orthogonal to the velocity of satellite v i , therefore the
last term in Eq. (4.29) is zero. Thus, in both the cases, the time derivative of the
Lyapunov function becomes:
N −1
N −1
V ( v, p) = −∑ v iT K i v i − ∑ ∇Oi 0 v 0 −
i =1
i =1
98
1
m0
N −1
∑m v κ
i =1
i
T
i
(4.30)
Since the linear momentum of the EM formation is conserved in the absence of external
perturbations, we have:
N −1
∑m v
i =1
i
i
= − m0 v 0
(4.31)
Using this relation, Eq. (4.30) can be written as:
N −1
N −1
i =1
i =1
V ( v, p) = −∑ v iT K i v i − ∑ ∇Oi 0 v 0 − vT0 κ
(4.32)
In order to make the Lyapunov function negative semi-definite, the parameter κ
is chosen according to the following relation:
N −1
κ = −∑ ∇OiT0
(4.33)
i =1
With this choice of κ , the time rate of change of the formation Lyapunov function
becomes:
N
V ( v, p) = −∑ v iT K i v i ≤ 0
(4.34)
i =1
Note that if the controlled system were not a switched system then Lyapunov
stability would follow from the negative semi-definiteness of Eq. (4.34). However, since
the controlled system is switched, Lyapunov stability does not follow from the negative
semi-definiteness of the individual Lyapunov functions (see Example 2.1 in [79]). To
prove stability for switched systems, an additional condition is required such that each
individual Lyapunov function is strictly monotonically non-increasing on the sequences
of all the switching times (see Theorem 2.3 in [79]). For the case of EMFF, the same
Lyapunov function is used for both instances of the switched controlled system, i.e. with
and without the obstruction maneuvering term. Equation (4.34) shows that this Lyapunov
function is negative semi-definite for all times and in particular at any switching times
that could occur. Therefore, the switched control system is stable in the sense of
99
Lyapunov. To show asymptotic stability for well-posed formations, the Invariance
Principle or LaSalle’s Theorem [30], [31], with additional conditions for asymptotic
stability of switched dynamical systems [81]-[82], will be used.
The basic argument in LaSalle’s theorem is that if the global minimum is
asymptotically stable then the formation cannot get “stuck” at any configuration other
than the global minimum due to the inherent dynamics and control law formulation. Let
x = [p v]T ∈
6N
be a point in the state-space X = {∀x ∈
6N
} . Then by construction
the Lyapunov function given by Eq. (4.19) is positive-definite in X. Define the set:
S = {x ∈ X | V ( v, p) = 0}
(4.35)
as the set of points where the time derivative of the Lyapunov function is zero. According
to LaSalle’s theorem, assuming that there are no accumulation points of switching times,
all solutions in the state-space converge to the largest invariant set in the set of points
where the time derivative of the Lyapunov function is zero, i.e. set S as defined above. To
determine the largest invariant set in S, it can be seen from Eq. (4.34) that:
V ( v, p ) = 0
⇒
v i (t ) = 0
⇒
v i (t ) = 0
∀i
(4.36)
From the switched feedback control law, given by Eqs. (4.25) and (4.26), zero
velocity for all times implies that:
N −1
(
)
M i pi + 2∑ ∇OijT − ∇OiT0 +
j =0
j ≠i
N −1
(
)
M i pi + 2∑ ∇OijT − ∇OiT0 +
j =0
j ≠i
mi
κ=0
m0
if pi ∉ Ti
mi
κ + γ i g imt = 0
m0
if pi ∈ Ti
(4.37)
(4.38)
By construction, Eq. (4.37) is never satisfied except when pi = 0 (which is the desired
equilibrium condition) otherwise if it is true for pi ≠ 0 then pi ∈ Ti ⊂ X which is a
100
contradiction. In the neighborhood around the local minima, i.e. set Ti , Eq. (4.38) is
applicable and no value of pi ∈ Ti ⊂ X can satisfy this equation due to the obstruction
maneuvering term γ i g imt which is added for this specific purpose. Thus, it can be seen
that the velocity of all the satellites cannot be zero simultaneously and in particular, in the
set Tk the velocity of the kth satellite would be nonzero, forcing the Lyapunov function to
be strictly negative-definite and consequently decreasing from one switching time to the
next. In summary due to the combination of the two facts namely,
1.
pi = 0 is the only zero velocity solution of Eq. (4.37),
2.
Lyapunov function is strictly decreasing in between any possible switching times
(see Theorem 2 in [81]),
the largest invariant set in the set S (Eq. (4.35)) is:
E = {p 0 }
(4.39)
hence the formation converges to this set asymptotically for almost all initial conditions.
In the above algorithm, it was assumed that the acceleration given by Eqs. (4.25)
and (4.26) can actually be realized by the EM actuation system. These equations are a set
of 3N-3 nonlinear polynomial equations in an unknown dipole strength vector μ . The
methods of solving this set of equations are discussed later in this chapter where it is
shown that these can be solved efficiently in real-time. Moreover, the electromagnetic
dipoles produce the desired acceleration by changing the currents in the coils. Since the
current in the superconductors is limited by the critical current density * resulting in
dipole saturation that has been neglected in the above proof. Nevertheless, theoretically
*
In a superconductor, if the current is increased beyond the critical current density then the property of
superconduction is lost.
101
speaking, the potential function of the formation can always be adjusted such that the
descent rate is sufficiently slow, resulting in small acceleration demand which can be met
by the dipoles without saturation. Hence asymptotic convergence of the formation can
always be achieved without saturation of the dipoles, at least in deep space where the
dipoles do not have to fight the gravitational terms. In LEO it is assumed that the dipoles
are designed such that they can provide sufficient acceleration required for the
reconfiguration and to cancel any gravitational disturbance terms. Therefore, the
saturation of the dipoles does not affect the stabilizability of the formation. Although
from a system design point of view, as discussed in Section 4.1.2, using APFM for
formation reconfiguration may not be practical, due to optimality considerations. Instead
optimal trajectories (see Chapter 5) or potential function shaping technique (Section 4.4)
can be used to improve convergence times.
4.3.1. Simulation Results
In order to test the algorithm developed in the last section for the reconfiguration of
general EM formations using APFM, a nonlinear simulation was built to simulate EM
formations in 2D. In all the results presented in this section, each satellite is assumed to
have same mass of 30 kg and each coil is assumed to have a value of 100 turns-m2.
First simulation result is presented for a five-satellite formation in which some of
the satellites are already present at their final positions. Despite this fact, as Figure 4.8
shows, they move aside to give way to other satellites in such a way that the formation
achieves its desired configuration. Figure 4.9 shows the Lyapunov function values as a
function of time for the formation and shows that although the Lyapunov function for an
102
individual satellite may increase during the maneuver, the total formation Lyapunov
function monotonically decreases for all times.
3
Sat. 3
* = start
2
y (m)
1
Sat. 0
Sat. 1
0
Sat. 2
-1
-2
Sat. 4
-3
-4
-3
-2
-1
0
x (m)
1
2
3
4
Figure 4.8: Trajectories for the five-satellite formation reconfiguration.
Lyapunov Function
0.06
sat 1
sat 2
sat 3
sat 4
total
0.05
V(t)
0.04
0.03
0.02
0.01
0
0
50
100
150
200
250
time (sec)
300
350
400
Figure 4.9: Lyapunov functions for the five-satellite formation.
103
450
The second simulation result is presented for a four-satellite square formation
with terminal conditions as mirror image of the initial conditions. The trajectories for the
formation are shown in Figure 4.10, which shows that the satellites initially descend
down the potential function and are repelled due to the obstruction functions and as a
result approach the neighborhoods of the local minima. Due to the presence of
obstruction maneuvering term in the controller, a circulation starts as an “emergent
behavior” and formation reconfigures itself while avoiding the local minima. The
Lyapunov function of the formation is given in Figure 4.11 and again shows that the total
formation Lyapunov function is monotonically decreasing for all times.
2
1.5
Sat. 0 traj.
Sat. 0 start
1
Sat. 1 start
y (m)
0.5
Sat. 1 traj.
0
Sat. 3 start
-0.5
-1
Sat. 2 start
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x (m)
0.5
1
1.5
2
2.5
Figure 4.10: Trajectories for the four-satellite square formation reconfiguration.
104
Lyapunov Function
0.2
sat 1
sat 2
sat 3
total
V(t)
0.15
0.1
0.05
0
0
50
100
150
time (sec)
200
250
300
Figure 4.11: Lyapunov function for the four-satellite square formation.
Lastly the simulation results for a random ten-satellite formation are presented.
The initial and terminal conditions for the formation were initialized using a uniformly
distributed random number generator in such a way that the distance between any points
in the initial and terminal conditions was greater than three times the coil diameter. These
initial and terminal conditions were adjusted such that the formation center of mass
remained at the origin. Using the control law given by Eqs. (4.25) and (4.26), the
resulting trajectories are shown in Figure 4.12.
In this simulation, each satellite was assumed to have a mass of 30 kg, nA = 100
turns-m2, a coil current saturation at 100 Amps, all diagonal entries of the K matrices =
1.5 while all non-diagonal entries = 0, all diagonal entries of the M matrices = 0.01 while
all non-diagonal entries = 0, and maneuver time was 750 sec. The obstruction functions
were constructed identically for all the satellites with a unity diagonal N matrix, λ1 = 10
and λ2 = 1 . The dipole equations were solved using the algorithm discussed in Section
4.5.2.
105
* = Initial points, circles = Terminal points
4
2
y (m)
0
-2
-4
-6
-8
-8
-6
-4
-2
0
2
x (m)
4
6
8
Figure 4.12: Collision-free trajectories for a 2D, ten-satellite random formation.
4.3.2. Computational Complexity of the Algorithm
In order to get an estimate of the computational complexity of the algorithm, a number of
random formations were simulated for different numbers of satellites in the formation.
The average simulation convergence time required to complete the maneuver is plotted
against the number of satellites in the formation as shown in Figure 4.13. This figure also
shows the average value of the time required for executing the simulation and shows that
the computational complexity increases almost linearly with increasing number of
satellites in the formation, at least for small N where N is the number of satellites in the
formation. These results were generated in Matlab on a 3.0 GHz desktop computer. Also,
106
the Matlab fmincon function was used to solve the dipole equations optimization problem
(discussed in Section 4.5). These results clearly highlight the fact that the given algorithm
can be used to generate feasible collision-free trajectories for quite complex
electromagnetic formations in real-time.
Simulation convergence times average (over 25 runs)
1000
800
sec
600
400
200
0
2
3
4
5
6
No. of sats in the random formation
7
8
7
8
Average computation times (over 25 runs)
1000
800
sec
600
400
200
0
2
3
4
5
6
No. of sats in the random formation
Figure 4.13: Average simulation and computation-time values for different number of
satellites in the random formations.
4.4. Potential Function Shaping for Reducing Maneuver Time
As discussed in Section 4.1.2, APFM generates sub-optimal trajectories which are not
suitable for EMFF. One way to improve the sub-optimality of the trajectories is by
reducing the maneuver time using Potential Function Shaping (PFS). The potential
function used up to this point was a quadratic with an exponential obstruction term. The
107
convergence rate for such a potential function is governed by the gradient of the quadratic
term (see Eq. (4.7)). This gradient approaches zero, as the EMFF satellites approach the
desired positions, due to which the satellites can spend considerable time at the end of the
maneuver as can be seen from Figure 4.2. This observation suggests a simple idea of
improving the convergence rates by adding additional terms to the potential function that
become significant near the desired final position. This idea will be explained further
with the help of the two satellite formation whose dynamics are given by Eq. (4.1).
Instead of using the potential function given by Eq. (4.3), the following potential
function is defined:
φ (t ) = 12 r1T Mr1 + O(r1 ) − γ 1 exp ( −γ 2 12 r1T Γr1 )
(4.40)
where γ 1 and γ 2 are positive shaping constants, and Γ is a positive definite shaping
matrix. Such a potential function is shown in Figure 4.14 (with exaggerated values for the
shaping gains). From the figure it is clear that the potential function is shaped so that its
gradient decreases much more slowly (as the satellite approaches the global minimum),
as compared to the quadratic PF.
The gradient of the potential function is given by:
(
)
∇φ (t ) = Mr1 + ∇O(r1 ) + γ 1γ 2 exp − 12 γ 2r1T Γr1 Γr1
(4.41)
This gradient can be written as:
∇φ (t ) = M eff (r1 )r1 + ∇O(r1 )
(4.42)
where M eff (r1 ) is the effective gain given by:
M eff (r1 ) = M + γ 1γ 2 exp ( − 12 γ 2r1T Γr1 ) Γ
108
(4.43)
Figure 4.14: A potential function for a 2-satellite formation in 2D with potential function
shaping added.
Thus for large values of the position error r1 , the effective gain approximately equals M
while near the global minimum the effective gain increases therefore increasing the
convergence rate.
It should be noted that the addition of the PF shaping term does not change the
Lyapunov stability of the formation, as can be seen from Eqs. (4.5) through (4.8), as long
as control inputs remain unsaturated. Using PF shaping, the new control law becomes:
r1 = − P −1 ( K (r1 , r1 )r1 + M eff (r1 )r1 + ∇OT )
(4.44)
The damping gain K (r1 , r1 ) can be adjusted along the trajectories so that the satellite
remains critically damped all the time. This leaves the designer with four parameters,
namely M , γ 1 , γ 2 and Γ , to adjust. With the help of a nonlinear simulation, these
parameters can be adjusted iteratively such that the maneuver time is as small as possible
without saturating the control inputs.
109
4.4.1. Two-Satellite Formation Simulation Results using PF Shaping
Consider a formation-flying telescope in deep-space with a detector module and a
primary module. This two-satellite formation needs to complete a 90° slew maneuver to
observe a new target. Using the nonlinear simulation with the PF shaping term added, the
simulation results are shown in Figure 4.15. For comparison, the same formation slew
maneuver was simulated without adding the PF shaping term and the results are shown in
Figure 4.16. It can be seen that using the PF shaping reduces the maneuver time from
over 1100 seconds to about 550 seconds without any control input saturation. Therefore,
from a practical point of view, the use of PF shaping reduces the suboptimality of the
trajectories considerably, making the algorithm more attractive for EMFF.
y1, y2 (dashed)
5
m
5
m
10
0
-5
0
0
0
-0.02
-0.04
0
500
1000
time sec
Ix2 and Iy2 (dashed)
0.04
50
50
0.02
A
m/s
100
A
100
0
-50
0
200
400
600
time sec
Meffx2 and Meffy2 (dashed)
1000
0
-50
500
1000
time sec
Kx2 and Ky2 (dashed)
0.06
0.04
500
0.02
0
500
1000
time sec
ydot1,ydot2 (dashed)
0
-0.02
0
0
500
time sec
1000
6
y (m)
x 1e-6
-5
200
400
600
time sec
xdot1, xdot2 (dashed)
0.02
m/s
x1, x2 (dashed)
10
4
2
0
-2
0
0
200
400
time sec
600
0
0
500
time sec
1000
-4 -2
0 2 4
x (m)
6
Figure 4.15: Two-satellite formation reconfiguration result with PF shaping.
110
8
y1, y2 (dashed)
5
m
5
m
10
0
0
-5
1000
2000
time sec
-100
A
A
0
1000
2000
time sec
Meffx2 and Meffy2 (dashed)
30
x 1e-6
0
0.01
10
0.005
0
1000
time sec
2000
1000
2000
time sec
ydot1,ydot2 (dashed)
0.02
0
0
0
1000
time sec
2000
0
0
-0.02
1000
2000
time sec
Kx2 and Ky2 (dashed)
0.015
20
0
0
0
-100
-0.01
-0.02
1000
2000
time sec
100
100
0
m/s
0
y (m)
-5
xdot1, xdot2 (dashed)
0.01
m/s
x1, x2 (dashed)
10
0
1000
time sec
2000
6
4
2
0
-2
-4 -2 0 2 4 6 8
x (m)
Figure 4.16: 2-satellite formation reconfiguration result without PF shaping.
4.5. Solving the Dipole Equations
In this section methods of solving the deipole equations for general EM formations will
be discussed. The control law for an EM formation is given in the form of a desired
acceleration that must be applied to each satellite in the formation at each control time
step. For example for the APFM, a control law can be written by combining Eqs. (4.25)
and (4.17) as follows:
N
⎧
m ⎫
fi (p, v i , μ) = −Pi −1 ⎨K i (pi , v i ) v i + M i pi + 2 ∑ ∇OijT (p i , p j ) − ∇OiT0 + i κ ⎬
m0 ⎭
j =1, j ≠ i
⎩
(4.45)
for i=1,…,N-1. Note the left hand side of the above equation is a nonlinear function of
the control input vector μ given by Eqs. (2.24) for LEO formations and Eqs. (2.64) for
deep-space 2D formations. Therefore, to find the desired control inputs, dynamic
111
inversion has to be used [38]. From this formulation, it is clear that the above control law
for general electromagnetic formations can be written in the following form:
f m (p, μ) = ac (p, v; c) + f d (p, v)
(4.46)
where:
fm :
3N
×
3N
→
3N
is the map for the magnetic accelerations on each satellite
ac :
3N
×
3N
→
3N
is the map for controller desired acceleration
fd :
3N
×
3N
→
3N
is the map for dynamics terms that are cancelled by the
controller
p = formation position vector
v = formation velocity vector
c = controller parameters
At each given state of the formation, Eq. (4.46) can be written as:
f m (p, μ) = a des
(4.47)
where:
a des = desired acceleration at current state (a constant vector)
Seen this way, Eq. (4.47) represents a set of 3N scalar equations in the unknown
vector μ of size 3N at each given state of the formation. Since EM actuation cannot
move the center of mass of the formation, as discussed in Chapter 1, there is another
constraint on the magnetic accelerations, namely:
N
∑mf
i =1
i mi
(p, μ) = 0
where mi is mass of the ith satellite in the formation, f m = [f m1 f m 2
(4.48)
f mN ]T is the
vector of magnetic accelerations acting on the satellites, and f mi is the specific magnetic
112
force on the ith satellite. This restriction means that the right hand side of Eq. (4.47) must
also satisfy:
N
∑m a
i =1
i des
=0
(4.49)
This will be ensured if the control law is formulated in such a way that it does not
try to move the center of mass of the formation. Using the above restriction, equations for
one of the satellites can be written in terms of those for the rest of the satellites in the
formation. In other words, the dynamics of one of the satellites would be dependent on
the dynamics of the rest of the formation, hence in an N-satellite EM formation there are
3N-3 translational degrees of freedom and not 3N. Therefore, Eq. (4.47) represents a set
of 3N-3 nonlinear equations with 3N unknowns. Since there are N − 1 vector equations
for N dipoles, essentially one of the dipoles is unconstrained and is termed a “free
dipole.”
Looking at the form of the force equations between two dipoles, given by Eq.
(2.20), it is clear that these equations are actually polynomial equations in the unknown
vector μ . There are essentially two ways in which these dipole equations can be solved
which are discussed in the following sections.
4.5.1. Solution by Fixing the Free Dipole
One way of solving the dipole equations is to fix the extra degree of freedom that is
present in the form of a free dipole and solve the equations using standard polynomial
equation solution techniques such as the continuation method or homotopy. See
Reference [10] for a detailed discussion of these methods.
113
4.5.2. Solution by Utilizing the Free Dipole for Angular Momentum Management
The free dipole gives extra degrees of freedom in the solution. For example, for each
“fixed” value of the free dipole there is at least one potential solution that exists for Eq.
(4.47), therefore there exists an infinite number of solutions to the dipole equations.
Utilizing these solutions to improve system performance is a desirable objective. As
discussed in Chapter 1, one of the undesired side effects of using dipoles is the shear
torque that acts on a satellite whenever a shear force is applied on it. This shear torque
needs to be countered by momentum storage devices and angular momentum
management can become an issue. Therefore, one of the ways in which the dipole
equations can be solved is to utilize the “free dipole” and select the one solution, out of
the possible infinite set of solutions, which results in minimum magnetic torque on all the
satellites in the formation. Reference [10] discusses various ways of achieving that, but in
this section a new algorithm for achieving this objective will be presented for which there
is no need to solve for all the solutions of the dipole equations.
In this new algorithm, instead of solving the dipole equations directly, the solution
of the equations is formulated as a nonlinear optimization problem. Incidentally, it should
be pointed out that many solution algorithms of nonlinear equations are essentially based
on nonlinear optimization tools [39]. Therefore, this algorithm from a mathematical point
of view is not very different from direct solution of the dipole equations except that it
utilizes the extra degree of the freedom of the free dipole to arrive at a better solution.
The Angular Momentum Management Optimization Problem (AMMOP) is defined as
follows:
114
N
{
min ∑ Tim (tk )
μ ( tk )
i =1
Subject to:
}
+ μTi (tk ) Wmi μ i (tk ) + [μ i (tk ) − μ i (tk −1 )] Wdi [μ i (tk ) − μi (tk −1 )]
T
W
f m (p, μ) = a des
N
∑mf
i =1
i mi
(4.50)
(p, μ) = 0
Where:
μ(tk ) = vector of unknown dipoles = [μ1
μ N ] at the current time step tk
T
μ(tk −1 ) = vector of dipoles values at the last time step tk −1
Wmi = diagonal dipole weighting matrix for satellite i
Wdi = diagonal dipole differential weighting matrix for satellite-i
i W = weighted 2 or infinity norm
Tim (tk ) = Magnetic torque acting on satellite i at time step tk
It can be seen from the cost function given in Eq. (4.50) that the optimization
problem is set-up in such a way that accomplishes three objectives:
1) It tries to minimize the total magnetic torque acting on each satellite in the formation.
2) It tries to distribute the dipole strengths “evenly” on each satellite in the formation.
Note that the weighting matrices, Wmi ’s, can be used to weight the dipoles on
different satellites according to the maximum dipole strengths they can achieve.
3) And lastly the new dipole vector is computed in such a way that its change from the
previously computed value is a minimum. This objective is added to avoid
unnecessary switching of the dipoles at different time steps. The reason that dipoles
may switch signs can be seen from Eq. (2.20) since changing the sign of both the
dipoles does not affect the magnitude and force between two dipoles. This sign
115
switching may be beneficial from an angular momentum point of view (see Chapter
3), but too rapid switching at each time step may make the solution unsuitable for
practical implementation. Therefore, this term can be used with a smaller or larger
weighting matrix, accordingly.
This solution methodology is used in the simulation results presented in this
thesis. The advantage of this methodology is that there is no need to find all solutions to
the polynomial dipole equations. Secondly, it ensures a continuous solution while at the
same time minimizing the magnetic torque on all satellites in the formation. Although it
may seem to be a complex problem, the cost is quadratic and once a solution is found by
the optimizer, the next solution is very near to this solution. Hence, initializing the
optimizer at each time step with the previously computed dipole vector, results in a very
rapid convergence to the desired solution. Therefore, this methodology is suitable for
real-time implementation for moderately sized (N~10) formations on the currently
available hardware.
4.6. Summary
In this chapter the Artificial Potential Function Method (APFM) was introduced with the
help of an example. The motion planning problem using APFM was also briefly
introduced, and its limitations from the point of view of EMFF were highlighted, while at
the same time some workarounds to those limitations were presented. By combining the
APFM and Lyapunov theory, a complete motion planning solution applicable to EMFF
for generating collision-free trajectories for general EM formations was presented. In
order to improve the formation maneuver times, the Potential Function Shaping method
was presented and, with the help of an example it was shown that the maneuver times are
116
reduced by almost a factor of two. Lastly methods for dynamic inversion of the dipole
equations were presented.
117
Chapter 5
Optimal Trajectory Generation for EM Formations
5.1. Introduction
Real-time motion planning is an active area of research especially for nonlinear and
constrained systems [36] like EMFF. For systems like EMFF, even finding feasible
trajectories can be a challenging problem. One such method using APFM was presented
in Section 4.1.2. It was pointed out that these methods generate trajectories which are not
optimal. For EMFF, the cost of using controls (current) is negligible; hence the formation
reconfiguration maneuvers are time optimal maneuvers. It is important to highlight that
using sub-optimal trajectories for formation reconfiguration using EMFF results in
underutilization of a designed system and hence can be regarded as a mass penalty during
the system design process. This is illustrated with the help of the following example.
5.1.1. An Illustrative Example Using the Gen-X Mission
The purpose of this example is to highlight the importance of using optimal trajectories
for EMFF and to take a look at the nature of these optimal trajectories using a practical
yet simple real-world application. Generation-X (Gen-X) is a concept space-based
telescope that will observe light from celestial objects in the X-ray wavelength band. The
science mission of the telescope requires sensitivity approximately 1000-times greater
than the current X-ray telescopes such as Chandra. This results in focal lengths greater
119
than 50 m and hence formation flying of the primary (housing optics) and detector
(housing sensors, etc.) modules of the telescope becomes a viable option. See Reference
[42] and references therein for further details about the mission. For the purpose of this
example it is assumed that the primary module has three orthogonal EM coils while the
detector module has only one coil (see Figure 5.1), while both the modules are assumed
to have three orthogonal reaction wheels for attitude control.
N
Coil y
primary
detector
c.m.
S
Coil z
(orthogonal coil)
z
y
N
S
N
x
S
Figure 5.1: Coil architecture for the Gen-X mission with block arrows showing the forces and
torques acting on the dipoles.
For scientific observations it is desired to slew the telescope in different
directions. To describe such slewing maneuvers in the xy-plane, the relative dynamics of
the detector module with respect to the primary module, in a cylindrical reference frame
attached to the center of mass of the primary module, can be described by the following
equations:
r − rθ 2 =
M D + M P 3μ0 μ Py μ D
M D M P 2π r 4
M + M P 3μ0 μ Pz μ D
rθ + 2rθ = D
M D M P 4π r 4
where:
(r ,θ ) = Polar coordinates of detector module w.r.t. primary c.m.
M P = Primary module mass
120
(5.1)
M D = Detector module mass
μ D = Detector module dipole strength
μ Py = Detector module y-dipole strength
μ Pz = Detector module z-dipole strength
In these equations it is assumed that during a slewing maneuver the primary and
detector attitude control keep the two modules facing each other. Using these dynamics a
simple non-optimal slewing profile can be defined as the “arc profile” in which the
detector moves along an arc at a fixed distance from the primary in the xy-plane. In such
a maneuver the y-coil on the primary module provides the necessary centripetal force
while the z-coil provides the necessary shear force. The results for a 90-degrees slew are
shown in Figure 5.2.
To determine the optimal slewing profile, an optimal control problem can be
posed as follows:
J = θ (t f ) → max;
t0 = 0; t f = T ; x0 = [ r0
0 θ0
0] ; r (t f ) = r0
T
(5.2)
In this problem, for a given coil mass for both the primary and detector modules, the total
slewing angle is maximized in a given time with the constraint that the final distance
between the detector and primary modules be the same as at the start, thus resulting in an
optimal slewing profile. This is a split boundary value problem that is difficult to solve
analytically, but by making the assumption that the detector coil carries constant
maximum current during the slewing maneuver, the problem is simplified since the
control inputs μ Py and μ Pz enter linearly into the dynamics. From Pontryagin’s
121
Minimum Principle (PMP) it is known that such systems result in “bang-bang” control
where the control input varies between its maximum and minimum allowed values [43].
Because of the symmetry of the problem, it can be seen that this results in a
control profile as shown in Figure 5.3. From this figure it is clear that only one parameter,
namely Tswitch, completely determines the optimal slew profile given the coil masses. This
parameter was determined by writing a Matlab code that performed a search over
switching times such that the conditions in Eq. (5.2) were satisfied subject to the
dynamics given by Eqs.(5.1). This is essentially a non-linear mathematical program that
can also be solved by using NLP (Non-Linear Program) solvers discussed later in this
chapter. In order to compare the results of the optimal profile with those of the arc
profile, the dipole mass, or equivalently the maximum dipole strength, was reduced such
that the same slewing angle was achieved in the same time as the arc profile. Using this
reduced coil mass, an optimal slewing profile is shown in Figure 5.4.
As can be seen by comparing Figure 5.2 and Figure 5.4, the control values
required for the optimal slewing profile are approximately half that required for the nonoptimal arc slewing profile. Therefore, using the optimal slewing profile results in nearly
a saving of half the coil mass as compared to the non-optimal profile; which is significant
from the system design point of view.
Another aspect of the optimal slewing profile or optimal EMFF trajectories in
general is that these trajectories are not necessarily minimum distance trajectories. As can
be seen from Figure 5.4, the detector module travels a much longer distance than the
straight line, to achieve time optimality. This is due to the fact that magnetic forces
between the dipoles become much stronger as the modules become closer, resulting in
122
much higher control authority and hence less time to complete the maneuver. This has
significant implications for any optimal trajectory planner for EMFF in the sense that the
planner cannot ignore the nonlinear nature of the dipole actuators. Any optimal trajectory
planning for EMFF must include the complete nonlinear dipole actuator dynamics,
otherwise the resulting trajectories will be highly suboptimal as demonstrated by the
above example.
r vs time
x 10
20
40
time min.
theta vs time
60
4
2
0
-2
0
20
60
x 10
1
0
-1
0
60
x 10
1
0
-1
0
m/s
m
51
50
49
0
rad/s
deg
100
50
5
x 10
40
mupy vs time
5
A-m2
A-m2
0
0
0
0
20
40
rdot vs time
-18
-3
20
40
time min.
thetadot vs time
60
20
40
time min
mupz vs time
60
20
40
time min
H vs time
60
20
40
time min
60
6
1000
Nms
100
50
0
49
49.5
50
r (m)
50.5
0
-1000
0
51
Figure 5.2: Non-optimal slew results for the Gen-X mission.
μ
Pz
A-m 2
A-m 2
theta (deg)
time min
r vs theta
T/2
T
μ
Py
time
time
Tswitch
T/2
T-Tswitch
T
Figure 5.3: Control profile for the optimal slew maneuver.
123
r
r vs time
dot
0.05
m/s
m
60
40
20
0
20
40
60
0
-0.05
0
time min.
theta vs time
rad/s
deg
100
50
0
0
5
0
-5
0
20
5
40
60
mupy vs time
A-m2
A-m2
x 10
20
40
60
x 10
4
2
0
0
x 10
5
0
-5
0
time min
r vs theta
100
5
40
time min.
thetadot vs time
60
20
60
40
time min
mupz vs time
20
40
60
40
60
400
50
0
20
-3
20
time min
H vs time
Nms
theta (deg)
vs time
30
40
r (m)
200
0
0
50
20
time min
Figure 5.4: Control profile for the optimal slew maneuver.
In summary, finding optimal trajectories for EMFF is important from a system
design point of view and is a nonlinear problem. The purpose of the following sections is
to formulate the optimal control problem for general electromagnetic formations and also
to present the algorithms that can be used to solve these problems.
5.2. Optimal Control Problem Formulation for EMFF
As demonstrated by the illustrative example in the last section, the optimal trajectory
planning problem for EMFF is coupled with the EMFF system design for any given
mission. Since the design of electromagnetic formation systems is beyond the scope of
this thesis, only the optimal trajectory planning for a given EMFF system will be
considered in this thesis; nevertheless the tools presented here can be extended to include
the overall optimal design process for general electromagnetic formations.
124
In the remainder of this chapter, Optimal Time Reconfiguration Problems (OTRP)
with collision avoidance for general electromagnetic formations will be considered. In
this problem the formation is to be reconfigured from a known initial state to a known
terminal state in minimum time. Formally, the OTRP for an N-satellite electromagnetic
formation can be defined as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
OTRP ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
tf
min J = ∫ dτ
0
μ ,t f
Subject to:
Dynamics:
Initial Conditions:
x = f (x, μ)
x0 = x(0) = [x10
Terminal Conditions: x f = x(t f ) = [x1 f
Control Constraints:
x N 0 ]T
x 20
x2 f
−μi max < μ i (t ) < μ i max
x Nf ]T
i = 1...N , t ∈ [0, t f ]
(5.3)
Collision Avoidance: min dij (t ) > Dmin ∀i ≠ j , i, j = 1...N , t ∈ [0, t f ]
RW Saturation:
hiRW (t ) < H i max
N
C.M. Constraint:
∑m x
i =1
where x :[0, t f ] →
Nns
i
i0
= ∑ mi xif
i =1
denotes the state vector of the formation, μ :[0, t f ] →
denotes the formation control vector, xi :[0, t f ] →
μ i :[0, t f ] →
nc
i = 1...N , t ∈ [0, t f ]
N
ns
Nnc
is the state vector of satellite i,
is the control vector for satellite i, Dmin is the minimum distance
allowed between any two satellites, H i max is the maximum angular momentum that can
be stored on the reaction wheels of satellite i, and dij (t ) represents the distance between
satellites i and j at time t. In this problem appropriate equations for the dynamics can be
used depending on the problem at hand, e.g. in LEO Eqs. (2.24) can be used while in
deep space Eqs. (2.55) may be used. Note that in this formulation the reaction wheel
angular momentum is managed as a hard constraint in the problem instead of minimizing
it in the cost integral. This formulation ensures that the reaction wheels never cross the
125
saturation limit and hence utilize the system to the maximum possible extent. The last
constraint in the problem is added to ensure that the terminal and initial conditions of the
formation are defined in such a way that the center of mass (C.M.) of the formation is not
moved.
5.3. Overview of Solution Methodologies
The optimal time control problem defined in the previous section is a coupled nonlinear
problem with control and trajectory constraints. There has been extensive research going
on for the solution of optimal control problems for the past half century or so and a large
bulk of literature exists on different methods for solving these problems [44]. The
solution approach for these problems can be roughly divided into two broad categories.
The first category invokes calculus of variations to derive the necessary optimality
conditions that define an optimal solution for the problem. These approaches, called
“indirect” methods, result in a coupled differential algebraic boundary value problem
(BVP). This BVP can be solved using a number of existing tools for solving such
equations such as shooting or multiple shooting methods [45]. The complication with this
approach is that this method is extremely sensitive to the distance of the current control
estimate from the optimal control. If we do not start our initial guess so that it is near to
the optimal value, then the chances of convergence are rather slim. Secondly, the
presence of trajectory and control constraints make the problem much harder to solve as
one needs to know the time intervals for constrained and unconstrained arcs in order to
solve the problem [46].
Another approach that will be used in this chapter is called a “direct” approach in
which the optimal control problem is transcribed into a finite dimensional mathematical
126
program and then this program is solved directly using standard packages. In this method
we can transcribe both the control and the state vectors and represent them in a finite
dimensional space and solve them using existing NLP packages. Another possible
implementation is to represent only the control vectors in a finite dimensional space and
use some fixed step explicit integration technique such as RK4 (Runge-Kutta order 4) or
higher order methods to integrate the dynamics from 0 to tf using the current estimate of
the control vector.
Figure 5.5: Relationship between “direct” and “indirect” methods and the Covector
Mapping theorem [47].
In order to understand the relationship between these different solution
methodologies, consider a basic optimal control Problem B (such as OTRP given in
(5.3)) and its solution paths as shown in Figure 5.5. In the “indirect” approach, Problem
Bλ is formed by using Pontryagin’s Minimum Principle (PMP) which itself needs to be
solved by approximating it to Problem Bλ N as discussed earlier. In the “direct” approach
127
the Problem B is directly discretized to Problem B N which is in general a nonlinear
program (NLP) and can be solved by using existing NLP packages. It is important to note
that using the Karush-Kuhn-Tucker (KKT) optimality conditions [39] for the Problem
B N give rise to the dual problem B N λ which, under appropriate conditions, can be
transformed to Problem Bλ N using the Covector Mapping Theorem [47]. This essentially
means that the costates of the optimal control problem can also be computed from the
Lagrange multipliers of the solution of the nonlinear program. It should be noted that not
all “direct” methods can be used to generate the costate estimates for the optimal control
problem this way, although the Legendre Pseudospectral method discussed later in the
chapter generates the costate vector by a very simple transformation of the Lagrange
multipliers of the NLP. For a historical perspective on the relationship of the direct and
indirect methods see [48].
5.4. Direct Shooting Method
In this section a direct shooting algorithm will be presented that will be used to solve
optimal time problem for EMFF, and the strengths and weaknesses of the algorithm will
be discussed.
5.4.1. A Description of the Algorithm
The direct shooting method, also sometimes known as the sequential method [49], the
control vector is parameterized by piecewise polynomials and optimization is done with
respect to these variables only. Given the initial conditions x0 , the optimal control
problem DAE (Differential Algebraic Equations) are solved using explicit integration
methods in a loop driven by the NLP solver as shown in Figure 5.6. A detailed
128
description of the method as it is applied to solve the OTRP (5.3) is given in the
following paragraphs.
û
x0
tf
NLP Solver
_
∫ f (x, uˆ )dt
+
0
xf
Figure 5.6: Direct shooting method for solving optimal control problems.
Since OTRP is a free terminal time problem, the free time variable is scaled
according to:
τ=
t
tf
(5.4)
With this change of variable, the OTRP can be written in the computational domain as
follows:
⎧
⎪
⎪
⎪
⎪
OTRPc ⎨
⎪
⎪
⎪
⎪
⎩
tf
min J = ∫ dτ
μ ,t f
Subject to:
Dynamics:
Initial Conditions:
0
x(τ ) = f (x, μ,τ )
x0 = x(0) = [x10 x 20
Terminal Conditions: x f = x(1) = [x1 f
Constraints:
c(x, μ) ≤ 0
x N 0 ]T
x2 f
(5.5)
x Nf ]T
τ ∈ [0,1]
where all of the constraints have been combined for simplicity and the function names
have been kept the same, although the independent variable is now different. Given the
129
initial conditions x0 , an estimate of the control vector μ̂ and terminal time t f , the
terminal solution to the dynamics can be computed as follows:
tf
x(1) = ∫ f (x, μˆ ,τ )dt
(5.6)
0
Since a closed form solution to Eq. (5.6) is not available in general (especially for
EMFF), the solution is approximated using an explicit integration method such as
classical RK4 (Runge-Kutta-4) [50] as follows:
x = xˆ (1) = x(0) +
M
f
h M −1
∑ k (xi , μˆ i , μˆ i +1 ,τ i )
6 i =0
(5.7)
where M is the number of intervals each of length h in which the computational domain
Nnx
[0,1] is divided for approximation and the function k :
×
Nnu
×
Nnu
×
→
Nnx
is
defined by the following relations:
k (xi , μˆ i , μˆ i +1 ,τ i ) = k1 + 2k 2 + 2k 3 + k 4
k1 = f (xi , μˆ i ,τ i )
k 2 = f ( xi + h2 k1i , 12 (μˆ i + μˆ i +1 ),τ i + h2 )
k 3 = f ( xi + k , (μˆ i + μˆ i +1 ),τ i +
h
2
1
2i 2
h
2
(5.8)
)
k 4 = f ( xi + hk 3i , μˆ i +1 ,τ i + h )
Note that in the above formulation, the value of the control vector at the middle of
the interval is approximated by taking its average. For better approximations the control
values at the middle of the intervals can also be defined as additional NLP variables. The
RK4 method gives results accurate to O(h 4 ) and can also be used to approximate the cost
integrals, if any, in the optimal control problem.
By defining a combined unknown control vector as μ M = [μˆ 0
μˆ 1
μˆ M ]T
where μˆ i is the unknown control vector at the grid point i, the above approximation of
the terminal conditions can be written as:
130
x Mf = RKINT ⎡⎣f ; x0 , μ M ) ⎤⎦
(5.9)
where the integration approximation is viewed as an operator RKINT acting on the vector
field f and taking initial conditions and a control vector as inputs. Viewed this way, it is
clear that for each given initial condition, and given control vector values at the grid
points, a trajectory is generated which will not be in general a feasible trajectory, i.e. it
will not satisfy all the constraints. To obtain a feasible and ultimately an optimal
trajectory, the following nonlinear optimization problem can be defined:
⎧
⎪
⎪
⎪
⎪
OTRP-M ⎨
⎪
⎪
⎪
⎪
⎩
min
J = g (t f , μ M )
M
μ ,t f
Subject to:
x f − x Mf = 0
x Mf = RKINT ⎡⎣f ; x0 , μ M ) ⎤⎦
(5.10)
c( x M , μ M ) ≤ 0
The NLP defined by OTRP-M fits the formation of standard NLP solvers such as
Matlab fmincon and SNOPT [51], [52]. These solvers use the SQP (Sequential Quadratic
Programming) technique that explicitly takes into account the equality and inequality
constraints by using an exact penalty to convert the problem into an unconstrained one,
and then solve a series of such problems to arrive at the local minima [39]. See Reference
[51] and references therein for further details about the SQP algorithm.
In the above formulation of the OTRP-M, the formation trajectory approximation
given by x M = [x0M
x1M
x MN ]T is needed to construct the nonlinear constraints
c(x M , μ M ) that can be generated by saving the output of RKINT at each grid point. Using
this trajectory, many nonlinear constraints such as collision avoidance, reaction wheel
saturation etc. can be implemented in this algorithm.
131
5.4.2. Some Theoretical Considerations and Limitations of the Algorithm
The direct shooting or sequential method essentially parameterizes an infinite
dimensional optimal control problem by approximating it in a finite dimensional space.
From a practical point of view, convergence of the algorithm at two levels must be
considered. First, the resulting NLP must converge to a solution and second, the solution
to the approximate problem must converge in some sense to the solution of the original
problem. It should be noted that the SQP algorithm is concerned with the convergence of
the NLP while the method that is used to discretize the optimal control problem must be
consistent in some sense such that it converges to the solution of the original problem as
the discretization level is made finer.
More specifically, consistent approximation in the sense of Polak [53] means that
the stationary points of the approximation problem converge to the stationary points of
the original optimal control problem as the step size of the RK method is decreased. It
should be noted that not all RK methods result in consistent approximations as shown in
[54]. A detailed theoretical discussion of the consistency of the approximation of direct
shooting algorithms is beyond the scope of this thesis; see References [53] and [54] for
further details.
For a successful implementation of the direct shooting algorithm, it is essential
that the underlying unknown control vector must be approximated by as few parameters
as possible. If a large number of parameters are required to parameterize the control
vector then the method suffers from at least two problems. One is that the computational
cost of numerically computing the gradients required by the NLP solver increases
132
exponentially with the number of unknown control parameters. Indeed the gradients of
the constraints can be approximated by the central difference as follows:
ci (μ M + δ μ j ) − ci (μ M − δ μ j )
⎡ ∂c ⎤
⎢ ∂μ M ⎥ =
2 δμ j
⎣
⎦i, j
(5.11)
where:
⎡ ∂c ⎤
th
⎢ ∂μ M ⎥ = (i,j) entry of the Jacobian matrix of constraint gradients
⎣
⎦i, j
δ μ j = A small vector change in the direction of μ Mj .
As can be seen from the formulation of the OTRP-M given in (5.10), the
computation of the ci’s requires integrating the dynamics once for each perturbation.
Thus the trajectory must be integrated at least n times where n equals the length of the
approximate control vector μ M . This cost can be reduced somewhat by recognizing that
perturbing a control variable later in time does not affect the initial time history of the
trajectory. Therefore, integration can start at a later point in time by using saved
trajectory information.
The second limitation of the direct shooting algorithm is the fact that the
computation of the gradients is more sensitive to the perturbations occurring earlier in the
trajectory, as these perturbations propagate to the end and can have very nonlinear effects
on the terminal constraints. This limitation can be overcome by using the so-called
“multiple shooting” algorithm in which the time is divided into segments and both
control and state variables are discretized at these segment end points, or cardinal nodes
[55], while only the control variables are discretized within each segment. This way the
errors due to gradient approximations remain within the segments and the algorithm
achieves better overall accuracy, although at the expense of a larger NLP. It should be
133
noted that in the limit when all the discretization nodes become cardinal nodes, the
algorithm is called “full transcription” in which both state and control variables are
discretized at all the nodes. One such algorithm using a Pseudospectral method is
discussed in Section 0.
5.4.3. Results
Consider the same problem given in the illustrative example of Section 4.1.1 where twosatellites switch places. The results using APFM are given in Figure 4.2 and it took over
750 seconds to reconfigure the formation. The optimal time results using the direct
shooting method are given in Figure 5.7.
x1 and x2 (x=2) vs time
y1, y2 (x=2) vs time
sat 1 & 2 (x) trajectory
2
5
1
2
0
0
0
-5
-1
-10
0
100
200
300
-2
0
x1dot, x2dot (x=2) vs time
0.2
y
10
-2
100
200
-5
300
y1dot, y2dot (x=2) vs time
0.1
0
x
10
Hrw1
Hrw2
5
0
-0.2
0
0
100
200
300
-0.1
0
ux optimal (x=2)
100
0
0
-100
-100
0
100
200
time (sec)
0
100
200
uy optimal (x=2)
100
0
5
300
-5
0
100
200
time (sec)
300
tf opt = 266.076053
M = 50
Max Total abs(Hrw) = 18.160496
Min Dist = 3.002609
100
200
time (sec)
Figure 5.7: Results for two-satellite that switch places using the direct shooting method.
134
The optimal time to complete the maneuver is 266 seconds for this highly constrained
and nonlinear problem. The maximum current, maximum reaction wheel angular
momentum and minimum distance were constrained to be 100 A, 10 N-ms and 3 m,
respectively.
5.5. Legendre Pseudospectral Method
Pseudospectral methods have gained popularity in the last five years due to the successful
application of these methods to trajectory generation for a large class of nonlinear
systems [56]-[60]. These methods are based on “full transcription” or discretization of
both the control and state variables of the optimal control problem at predefined time
values or nodes. The underlying unknown control and state vectors are approximated by
interpolating their values at the nodes using orthogonal polynomials as discussed in the
following section.
It is important to point out that Pseudospectral methods are based on representing
the unknown trajectory using global polynomials. Therefore, if the trajectory and/or
control history has discontinuities, then approximation errors will be larger, and in such
cases generally the trajectory is broken into phases or knots [61] and represented using
piecewise continuous polynomials. For EMFF in general, the trajectory and control
histories are discontinuous (see Figure 5.7) due to constraints in the system and the
minimum time nature of the problem. Therefore, to find time optimal trajectories, these
phases have to be defined. Unfortunately, the number of these phases is not known a
priori for a given problem and, moreover, these phases can increase the complexity and
computation time of the solution. Another disadvantage, from a practical point of view, is
that the presence of switching in the control inputs can make the actual real-time
135
implementation of these optimal trajectories difficult. To avoid these difficulties, the
optimal control problem for EMFF OTRP (5.3) will be modified so that it always
generates at least C1-solutions (i.e. at least one-time differentiable):
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
OTRPC ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min J = ∫
0
μ ,t f
Subject to:
Dynamics:
Initial Conditions:
tf
(1 + μ Rμ ) dτ
T
x = f (x, μ)
x0 = x(0) = [x10
Terminal Conditions: x f = x(t f ) = [x1 f
x N 0 ]T
x 20
x Nf ]T
x2 f
−μi max < μ i (t ) < μ i max
i = 1...N , t ∈ [0, t f ]
(5.12)
Collision Avoidance: min dij (t ) > Dmin ∀i ≠ j , i, j = 1...N , t ∈ [0, t f ]
RW Saturation:
hiRW (t ) < H i max
hiRW (t f ) = H if
RW Terminal Constraint:
C.M. Constraint:
N
∑m x
i =1
i = 1...N , t ∈ [0, t f ]
i
i = 1, 2,
,N
N
i0
= ∑ mi xif
i =1
where μ represents the time derivative of the control vector and R is a weighting matrix.
This formulation minimizes the 2-norm of the rate of change of the control vector and
hence forces the solution to be at least C1-continuous.
In the following section an algorithm will be developed using Legendre
polynomials that will be suitable for solving a general EMFF optimal control problem
such as OTRPC (5.12).
5.5.1. Approximating Functions, their Derivatives and Integrals using Orthogonal
Polynomials
An arbitrary function y (t ) can be approximated by an Nth degree polynomial at the N+1
interpolation points using the Lagrange interpolation formula [64] as follows:
N
y (t ) ≈ y (t ) N = ∑ y (ti )φi (t )
i =0
136
−1 ≤ t ≤ 1
(5.13)
where ti ’s are the discretization time values, and φi (t ) is the ith N+1 degree interpolating
basis polynomial (also called Cardinal Function) and is defined as follows:
t − tm
m = 0 ti − t m
N
φi (t ) = ∏
(5.14)
m ≠i
Note that φi (t ) ’s are generally constructed using Chebyshev or Legendre
polynomials. The interpolation points (also called collocation points) are defined by their
corresponding roots in order to avoid the Runge Phenomenon [60] and as well as to get
the minimum possible approximation error. Let,
N
w(t ) = ∏ (t − tm )
(5.15)
m=0
be the N+1 degree polynomial with N+1 roots {t0 , t1 ,
t N } . Then, its derivative at
the roots can be written as:
N
w(ti ) = ∏ (ti − tm )
m =0
m ≠i
(5.16)
Using these definitions, the Cardinal Function can be written as:
N
w(t )
m = 0 (t − ti ) w(ti )
φi (t ) = ∏
(5.17)
m ≠i
Note that a desired property of Cardinal Functions, that can be seen by an inspection of
Eq. (5.17), is that they are orthonormal, i.e.:
⎧1
⎩0
φi (tk ) = ⎨
if i = k
otherwise
(5.18)
It is known that interpolating the function at Gauss quadrature points results in
minimum approximation error to the function and its derivatives [60]. For BVPs
(Boundary Value Problems), for which both the initial and terminal conditions are
defined, LGL (Legendre-Gauss-Lobatto) points defined by:
137
t0 = −1
LN (tk ) = 0
k = 1, 2,
, N −1
(5.19)
tN = 1
form the nodes of Legendre Cardinal Functions. Here, LN (t ) is the derivative of the Nth
degree Legendre polynomial. Legendre polynomials are orthogonal over the interval
[−1,1] , i.e.,
N≠M
⎧ 0
⎪
∫ LN (t ) LM (t )dt = ⎨ 2 N = M
−1
⎪⎩ 2 N + 1
1
(5.20)
and are solutions to the following second order differential equation [77]:
(1 − t 2 ) LN (t ) − 2tLN (t ) + N ( N + 1) LN (t ) = 0
(5.21)
Using LGL-points, Eq. (5.15) can be written as:
w(t ) = (t 2 − 1) LN (t )
(5.22)
Similarly using Eq. (5.21), Eq. (5.16) can be written as:
w(tk ) = N ( N + 1) LN (tk )
(5.23)
Using Eqs. (5.22) and (5.23), the Cardinal Function given by Eq. (5.17) can be written as:
φi (t ) =
(t 2 − 1) LN (t )
N ( N + 1)(t − ti ) LN (tk )
(5.24)
Note that the Cardinal Functions defined by the above equation are called
Legendre Cardinal Functions since Legendre polynomials are used in their construction.
Using a similar procedure, other Cardinal Functions can be constructed using, for
example, Chebyshev polynomials. Legendre Cardinal Functions are shown for N=5 in
Figure 5.8 which clearly shows the orthonormality of these functions and uneven
distribution of the LGL-points.
138
Using Eq. (5.13), the derivative of the unknown function y (t ) can be
approximated by:
N
y (t ) ≈ y (t ) N = ∑ y (ti )φi (t )
−1 ≤ t ≤ 1
(5.25)
i =0
Therefore, the derivative can be simply approximated by evaluating the derivatives of the
Cardinal Functions. More specifically, at the collocation points:
N
y (tk ) N = ∑ y (ti )φi (tk )
(5.26)
i =0
By defining a vector: y N = [ y (t0 )
y (t N )]T , the above equation can be written
y (t1 )
as:
y N = Dy N
(5.27)
where D is an ( N + 1) × ( N + 1) matrix.
Figure 5.8: Cardinal functions (N=5) using Legendre Polynomials and LGL points
(squares)
139
The matrix D can be computed by differentiating Eq. (5.24) and, after some
algebraic manipulations and using some properties of Legendre polynomials, can be
shown to be:
⎧ LN (tk ) 1
⎪ L (t ) t − t
⎪ N i k i
⎪ − N ( N + 1)
⎪
D = [ Dki ] = ⎨
4
⎪ N ( N + 1)
⎪
4
⎪
⎪⎩
0
k ≠i
k =i=0
(5.28)
k =i= N
otherwise
In a similar fashion the integral of a function can be approximated as:
t
t
∫ y(τ )dτ ≈ ∫ y(τ )
−1
−1
N
N
t
i =0
−1
dτ = ∑ y (ti ) ∫ φi (τ )dτ
−1 ≤ t ≤ 1
(5.29)
The values of the integral at the collocation points can be written as:
tk
∫
−1
where W ∈
N ×( N +1)
N
y (τ ) N dτ = ∑ y (ti ) Wki
(5.30)
i =0
is an integration matrix whose entries are defined by:
tk
Wki = ∫ φi (τ )dτ
(5.31)
−1
It can be shown that the entries of this matrix are given by [65]:
Wki =
ωk ⎡
N
⎤
1
+
t
+
Lm (tk ) { Lm +1 (ti ) − Lm −1 (ti )}⎥
∑
i
⎢
2 ⎣
m =1
⎦
i = 0,1, , N
k = 1, 2, , N
(5.32)
where ωk ’s are familiar Gauss Quadrature weights that are used to evaluate the integral
of an unknown function from -1 to 1 and are given by:
1
ωk = ∫ φk (τ )dτ =
−1
2
1
N ( N + 1) LN (tk ) 2
140
(5.33)
Note that these approximations would be exact (i.e. error would be zero) if the
unknown function y (t ) were a polynomial of degree N-1 or less. For other functions the
Cauchy Interpolation Error Theorem [60] gives the error as follows:
1
d N +1 y (t )
(ξ ) w(t )
y (t ) − y (t ) =
( N + 1)! dt N +1
N
(5.34)
where y (t ) is assumed to be at least N+1 times differentiable, i.e. y (t ) ∈ C N +1 ,
ξ ∈ [−1,1] , and w(t ) is defined by Eq. (5.15) and Eq. (5.22) for Legendre Cardinal
Functions. By integrating and differentiating Eq. (5.13), the error bounds for the
derivative and integral approximations can be computed. It is important to see that for a
given interpolation scheme, such as at LGL points, the error is a function of the
smoothness of the unknown function. This is demonstrated by plotting the infinity norm
of the errors of two test functions defined as follows:
1
t +2
1
f 2 (t ) = 2
t +2
f1 (t ) =
4
−1 ≤ t ≤ 1
(5.35)
−1 ≤ t ≤ 1
These functions and their derivatives were approximated by the Legendre
Cardinal Functions and the infinity norm of the errors is plotted in Figure 5.9. This plot
clearly shows that the error is reduced by increasing the degree of the polynomial;
moreover, the effect of the smoothness of the function is also clearly visible as the
interpolation error for the “smoother” function f 2 is smaller. The effect of the
smoothness of the function can be seen by looking at the infinity norm of the nthderivative of the functions. Let,
141
g1 (n) =
1
d n f1 (t )
( N + 1)! dt n
− 1 ≤ t ≤ 1,
∞
n
d f 2 (t )
1
g 2 ( n) =
( N + 1)! dt n
(5.36)
n ∈ {1, 2, }
∞
then, g1 and g 2 give an estimate of the interpolation error due to the inherent nature of
the underlying function (scaled by 1/( N + 1)! ). This result is shown in Figure 5.10 and
clearly shows that a smoother function (e.g. f 2 ) results in much less interpolation error as
compared to a more rapidly changing function such as f1 .
2
10
f1 error norm
f2 error norm
0
10
f1dot error norm
f1dotdot error norm
f2dot error norm
-2
10log(||error||)
10
f2dotdot error norm
-4
10
-6
10
-8
10
-10
10
-12
10
5
10
15
Polynomial degree N
20
25
Figure 5.9: Infinity norm of the interpolation error for test functions given by Eq. (5.35)
142
0
10
-1
10
g1
g2
-2
10
-3
10
-4
10
-5
10
0
2
4
6
8
10
12
Derivative order
14
16
18
20
Figure 5.10: Infinity norm of the derivative for test functions given by Eq. (5.35)
5.5.2. Review of the Legendre Pseudospectral Method
In this section a Legendre Pseudospectral algorithm [62], [63] will be developed that is
applicable to fairly general optimal control problems and in particular to the OTRPC
(5.12). Consider the following control problem:
⎧
⎪
⎪
⎪
⎪
⎪
⎪⎪
G⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩
min J = h ( x(τ f ),τ f ) + ∫ g ( p(τ ), v (τ ), u(τ ), u(τ ) ) dτ
τf
τ0
u ,τ f
Subject to:
Dynamics:
Initial Conditions:
p(τ ) = v (τ )
v (τ ) = f (x, u,τ )
ψ 0 ( x(τ 0 ),τ 0 ) = 0
Terminal Conditions:
ψ f ( x(τ f ),τ f ) = 0
Algebraic Constraints:
c ( x(τ ), u(τ ),τ ) ≤ 0
τ
Integral Constraints:
τ ∈ [τ 0 ,τ f ]
∫ q ( x(s), u(s), s )ds − z(τ ) ≤ 0
τ0
143
(5.37)
τ ∈ [τ 0 ,τ f ]
τ ∈ [τ 0 ,τ f ]
In the general problem G, the objective is to determine the control function u(τ ) ∈
and state trajectory x(τ ) = [p(τ ) p(τ ) ] ∈
T
given above. The functions ψ 0 ∈
and terminal constraints and c ∈
p
r
2n
and ψ f ∈
m
that minimizes the Bolza cost function
q
, with p, q ≤ 2n , represent the initial
are nonlinear constraints on both the state and control
vectors. The integral inequality constraints are added to enforce generalized constraints
such as the reaction wheel angular momentum constraints required in the problem
OTRPC (5.12).
Note that the dynamics in this formulation are specifically applicable to
Newtonian translational dynamics of mechanical systems such as EMFF. The reason for
representing the dynamics in this form is that the Legendre Pseudospectral method can be
used to save the number of unknown NLP variables by exploiting higher derivatives as
discussed in the following paragraphs.
The problem G is defined over an interval [τ 0 ,τ f ] while the orthogonal
polynomials are defined over [-1,1]. Therefore the problem is transformed to the
computational domain using the following transformation:
τ = 12 (τ f − τ 0 ) t + 12 (τ f + τ 0 )
Using this transformation, the problem G transforms to:
144
(5.38)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Gc ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min J = h ( x(1),τ f ) +
τ f −τ 0
2
u ,τ f
Subject to:
p(t ) =
Dynamics:
v (t ) =
1
∫ g ( p(t ), v(t ), u(t ), u(t ) ) dt
−1
τ f −τ 0
2
τ f −τ 0
v (t )
t ∈ [-1,1]
f ( x, u , t )
2
ψ 0 ( x( −1),τ 0 ) = 0
Initial Conditions:
(5.39)
Terminal Conditions: ψ f ( x(1),τ f ) = 0
Algebraic Constraints: c ( x(t ), u(t ), t ) ≤ 0
Integral Constraints:
τ f −τ 0
2
t ∈ [−1,1]
t
∫ q ( x(s), u(s), s )ds − z(t ) ≤ 0
t ∈ [-1,1]
−1
Note that in problem Gc the variable and function names are kept the same for simplicity.
As discussed previously, the Legendre Pseudospectral method discretizes the problem by
representing the unknown functions as follows:
N
p(t ) ≈ p(t ) N = ∑ p(ti )φi (t )
i =0
N
u(t ) ≈ u(t ) = ∑ u(ti )φi (t )
N
(5.40)
i =0
where the Cardinal Functions φi (t ) are defined by Eq. (5.24).
Substituting these approximations into the cost integral in problem Gc results in
the following approximation cost:
J N = h ( x(1) N ,τ f ) +
τ f −τ 0
2
1
∫ g ( p(t )
N
, v (t ) N , u(t ) N , u(t ) N ) dt
(5.41)
−1
Using Gauss-Lobatto integration [66] the cost can be further approximated as:
J N = h ( x(1) N ,τ f ) +
τ f −τ 0
2
N
∑ g ( p(t )
k =0
N
k
145
, v (tk ) N , u(tk ) N , u(tk ) N ) ⋅ ωk
(5.42)
where ωk ’s are Gauss-Quadrature weights given by Eq. (5.33). It should be clear that by
construction of the interpolation, p(tk ) N = p(tk ) and u(tk ) N = u(tk ) . For notation
simplification let p(tk ) = p k and u(tk ) = u k which represent the values of the position
state and control vectors at the LGL (Legendre Gauss Lobatto) grid points, or collocation
points. Note that each position vector sample is of size n while each control vector
sample is of size m, i.e. x k ∈
n
and u k ∈
m
.
Combining these samples, the full unknown state values are defined in the form of
a matrix as follows:
⎡ pT0 ⎤ ⎡ p01
⎢ T⎥ ⎢
p
p
N
P = ⎢ 1 ⎥ = ⎢ 11
⎢ ⎥ ⎢
⎢ T⎥ ⎢
⎣⎢p N ⎦⎥ ⎣ pN 1
Note that P N ∈
( N +1)× n
p02
p12
pN 2
p0 n ⎤
p1n ⎥⎥
⎥
⎥
pNn ⎦
(5.43)
, thus each column of this matrix represents the samples of a given
position state at the collocation points. Similarly the control matrix can be defined as:
⎡ uT0 ⎤ ⎡ u01 u02
⎢ T⎥ ⎢
u
u12
u
N
U = ⎢ 1 ⎥ = ⎢ 11
⎢ ⎥ ⎢
⎢ T⎥ ⎢
⎢⎣u N ⎥⎦ ⎣u N 1 u N 2
Note that U N ∈
( N +1)× m
u0 m ⎤
u1m ⎥⎥
⎥
⎥
u Nm ⎦
(5.44)
. In this matrix notation the superscript N is included to remind us
that these vectors are the samples of the unknown state trajectory and control histories in
the space of Nth order polynomials, i.e. p(t ) N , u(t ) N ⊆ P N . Given this notation, the
velocity can be computed simply by using the differentiation matrix given by Eq. (5.28)
as follows:
V N = DP N
146
(5.45)
( N +1)× n
where V N ∈
. In this formulation, since v (t ) N is completely determined once
p(t ) N is known, there is no need to define the matrix V N as unknown variables in the
discretization of problem Gc resulting in considerable savings in the number of NLP
variables. In a similar fashion the time derivative of the control can be computed as:
U N = DU N
( N +1)×m
where U N ∈
(5.46)
is a matrix that gives the rate of change of each control input at each
collocation point and is defined as:
⎡ uT0 ⎤ ⎡ u01 u02
⎢ T⎥ ⎢
u
u12
u
N
U = ⎢ 1 ⎥ = ⎢ 11
⎢ ⎥ ⎢
⎢ T⎥ ⎢
⎣⎢u N ⎦⎥ ⎣u N 1 u N 2
u0 m ⎤
u1m ⎥⎥
⎥
⎥
u Nm ⎦
(5.47)
Similar to the cost integral, the dynamic equations in problem Gc are
approximated as:
2
⎛ τ −τ 0 ⎞
N
N
N
p(t ) = ⎜ f
⎟ f ( p(t ) , p(t ) , u(t ) , t )
2
⎝
⎠
N
(5.48)
These equations represent the dynamic constraints on the variables and can be enforced
by requiring that they be satisfied at the collocation points. This is the reason that these
methods are sometimes called “collocation” methods. Exact satisfaction of Eq. (5.48) at
the collocation points translates into the following nonlinear equality constraints:
2
⎛ τ −τ 0 ⎞
N
N
N
p(tk ) = ⎜ f
⎟ f ( p(tk ) , p(tk ) , u(tk ) , tk )
⎝ 2 ⎠
N
k ∈ {0,1,
, N}
(5.49)
Using the matrix notation defined above, these constraints can be written compactly as:
2
⎛ τ −τ 0 ⎞ N
D P −⎜ f
⎟ F =0
⎝ 2 ⎠
2
N
147
(5.50)
where D2 represents the square of the differentiation matrix used to obtain the second
derivative. The dynamics matrix F N ∈
( N +1)× n
, approximates the dynamics vector field in
the space of Nth order polynomials, as defined by:
⎡f0T ⎤ ⎡ f 01
⎢ T⎥ ⎢
f
f
N
F = ⎢ 1 ⎥ = ⎢ 11
⎢ ⎥ ⎢
⎢ T⎥ ⎢
⎢⎣f N ⎥⎦ ⎣ f N 1
where fi = f ( p(ti ) N , p(ti ) N , u(ti ) N , ti ) ∈
f 02
f12
fN 2
n
f0n ⎤
f1n ⎥⎥
⎥
⎥
f Nn ⎦
(5.51)
. Note that matrix Eq. (5.50) represents
( N + 1) × n nonlinear constraints which are functions of the unknown variables defined by
the matrices P N and U N .
In the same vein, the integral constraints are collocated, i.e. enforced at the
collocation points as follows:
τ f −τ 0
2
tk
∫ q ( x( s )
N
, u( s ) N , s )ds − z (tk ) ≤ 0
∀k ∈ {1, 2,
, N}
(5.52)
−1
The integral in the above constraints can be further approximated by using the integration
matrix given by Eq. (5.32) and the constraints can be written as:
τ f −τ 0
2
N
∑ q ( x(t )
i =0
i
N
, u(ti ), ti ) ⋅ Wki − z (tk ) ≤ 0
∀k ∈ {1,
, N}
(5.53)
Using matrix notation these constraints can be compactly written as:
⎛ τ f −τ 0 ⎞
N
⎜
⎟ WQ − Z ≤ 0
⎝ 2 ⎠
The matrix Q N ∈
( N +1)×l
is defined as:
148
(5.54)
⎡ qT0 ⎤ ⎡ q01
⎢ T⎥ ⎢
q
q
N
Q = ⎢ 1 ⎥ = ⎢ 11
⎢ ⎥ ⎢
⎢ T⎥ ⎢
⎣⎢q N ⎦⎥ ⎣ qN 1
where qi = q ( p(ti ) N , p(ti ) N , u(ti ) N , ti ) ∈
q0l ⎤
q1l ⎥⎥
⎥
⎥
qNl ⎦
q02
q12
qN 2
l
(5.55)
and the given matrix Z ∈
⎡ z (t0 )T ⎤ ⎡ z01
⎢
⎥ ⎢
z (t1 )T ⎥ ⎢ z11
⎢
=
Z=
⎢
⎥ ⎢
⎢
⎢
T⎥
⎣⎢ z (t N ) ⎦⎥ ⎣ z N 1
N ×l
is defined as:
z0 l ⎤
z1l ⎥⎥
⎥
⎥
z Nl ⎦
z02
z12
zN 2
(5.56)
Using the above relations, the problem Gc (5.39) is completely discretized and can
be written as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪⎪
G cN ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩
min
J N = h ( p N , v N ,τ f ) +
N
N
τ f −τ 0
P , U ,τ f
2
N
∑ g (p , v
k =0
k
k
, u k , u k ) ⋅ ωk
Subject to:
2
⎛ τ −τ 0 ⎞ N
D P −⎜ f
⎟ F =0
⎝ 2 ⎠
ψ 0 ( p 0 , v 0 ,τ 0 ) = 0
2
Dynamics:
Initial Conditions:
N
(5.57)
Terminal Conditions: ψ f ( p N , v N ,τ f ) = 0
Algebraic Constraints: c ( p k , v k , u k , tk ) ≤ 0
∀k ∈ {0,1,
, N}
⎛ τ −τ 0 ⎞
N
Integral Constraints: ⎜ f
⎟ WQ − Z ≤ 0
⎝ 2 ⎠
Note that in the problem G cN , the unknowns are P N ∈
terminal time τ f ∈
( N +1)× n
, UN ∈
( N +1)×m
, and the
. Therefore there are a total of ( N + 1)(n + m) + 1 unknown variables.
Using this formulation, i.e. direct use of the second derivative in the dynamics instead of
the state space representation, there has been a savings of ( N + 1)n variables which is
significant for large electromagnetic formations. The above problem is a NLP (Nonlinear
Program) that can be solved by standard NLP packages as discussed in Section 5.3.
149
5.5.3. Results
For a four-satellite square formation, a set of optimal trajectories was generated for 90°
CCW (Counter Clock Wise) rotation using the LPS method with a grid size of N=25. The
results are shown in Figure 5.11. These results show the optimal trajectories as well as
the control currents and the reaction wheel angular momentum storage required for the
maneuver. As can be seen from the figure, the optimal terminal time is around 341
seconds and the reaction wheel angular momenta, which started at zero, return to zero at
the terminal state. Moreover, the optimal trajectories are characteristic of EMFF in that
the satellites try to approach each other in order to maximize the control authority to
execute a minimum time maneuver.
ix2 (*), iy2 (circle)
100
100
100
0
0
0
-100
0
200
time sec
-200
400
-100
0
200
time sec
Hrw1 vs time
-200
400
5
5
100
0
0
0
-100
-200
Nms
200
Nms
A
-200
A
200
-100
-5
-10
0
200
-15
400
200
time sec
Hrw2 vs time
400
0
200
400
-5
0
200
-15
400
Hrw4 vs time
0
0
-5
-10
trajectories
sat 1
5
y (m)
5
Nms
5
-15
0
-10
Hrw3 vs time
Nms
200
A
A
200
-5
sat 4
0
sat 2
-10
0
200
time sec
400
-15
-5
0
200
time sec
400
sat 3
-5
0
x (m)
5
Figure 5.11: Optimal trajectory results using the LPS method for a four-satellite
formation 90° CCW rotation.
150
The second result is presented for a random ten-satellite formation. For this same
formation, the collision-free trajectories were generated using APFM as shown in Figure
4.12. These trajectories were used as the initial starting guess for the LPS method (with
N=25). The resulting optimal trajectories are shown in Figure 5.12. The optimal time to
execute the maneuver was 134 seconds while the APFM took around 800 seconds to
complete the maneuver. For a comparison of the APFM and the Optimal Trajectory
Following Method (OTFM) see Section 6.4.
10 sat. random formation optimal trajectories (start=squares)
8
6
4
y (m)
2
0
-2
-4
-6
-8
-10
-5
0
x (m)
5
10
Figure 5.12: A ten-satellite random formation optimal trajectories results using LPS
method.
151
Lastly, the optimal trajectories for a two-satellite 90° slew maneuver are
presented. The trajectories for this formation were generated using APFM and, for
comparison, the results are given in Figure 4.15. The optimal trajectory results for N=55
are shown in Figure 5.13. Note that these trajectories were generated with the reaction
wheel angular momentum constraint of 3 Nms. Even with this constraint, the maneuver
took 330 seconds as compared to the APFM result of around 560 seconds.
rx1 (*), ry1 vs time
vx1 (*), vy1 vs time
10
2
0.1
0
0
Nms
m
m/s
5
-5
Hrw0 (*), Hrw1 vs time
0.2
0
-2
-0.1
0
200
time sec
ix1 (*), ix2 vs time
-0.2
400
0
200
time sec
iy1 (*), iy2 vs time
-4
400
200
300
400
time sec
traj. sat0 (dashed) and sat1 (*=start)
200
200
100
100
4
0
0
2
0
100
-100
-100
-200
-200
m
A
A
6
0
-2
0
200
time sec
400
0
200
time sec
400
-4
-2
0
2
m
4
6
8
Figure 5.13: Trajectories generated using LPS method with N=55 for a two-satellite 90°
slew maneuver.
5.6. Summary
In this chapter the optimal trajectory generation problem for general EM formations was
presented. It was shown that the optimal trajectory problem for EMFF is a nonlinear and
constrained problem due to the inherent dynamics of EMFF. Different solution
methodologies and their inter-relationships were also discussed briefly. Two “direct”
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solution algorithms, namely the Direct Shooting method and the Legendre Pseudospectral
(LPS) method, were presented for EMFF. Results using example EM formations were
also presented for both methods. The Direct Shooting method is suitable if the optimal
control problem can be parameterized using only a few variables, whereas the LPS
method is based on full-transcription of the optimal control problem and is suitable for
large, highly constrained and nonlinear problems. The LPS method results in a sparse
NLP that can be solved efficiently using existing NLP-solvers such as SNOPT.
153
Chapter 6
Real-Time Implementation of Optimal Trajectories
6.1. Introduction
Optimal trajectory generation for nonlinear and constrained systems, such as EMFF, is
computationally expensive as discussed in the last chapter. Due to the complexity of the
dynamics and constraints, we have to resort to numerical algorithms that generate “openloop” trajectories and control histories that not only take a lot of computational power
and time but also cannot be used directly for implementation on actual systems. Even in a
perfect simulation in the Matlab environment, the system will not follow these
trajectories and diverge, if we apply the pre-computed controls, due to small numerical
errors. In a realistic environment, there are always modeling errors and external
unmodelled disturbances. Therefore, a “closed-loop” scheme is needed in order to
actually implement these optimal trajectories on real-world systems.
The purpose of this chapter is to provide two practical methods that can be used to
implement the optimal trajectories for nonlinear and constrained systems in real time with
relatively low computational burden.
6.2. Adaptive Trajectory Following Formulation
A simple method with low computational burden is to use the adaptive trajectory
following formulation for EMFF discussed in Chapter 3. Since time optimal trajectories
155
for EMFF result in controls operating near or at constraint boundaries, trajectory
following control laws, as developed in Chapter 3, cannot be used directly. This is
apparent since once the controls are saturated at the boundaries, it can change only in one
direction and hence only provide the ability to steer the system by reduction in controls
when faced with external disturbances and uncertainties in the model. The full control
authority can be recovered by the method of “constraint tightening,” in which the optimal
trajectories are computed offline by reducing the constraint margins. This reduction in
constraint margins can be done according to the size of unknown but bounded
disturbances as discussed below.
6.2.1. An Upper Bound for Constraint Tightening for Bounded Disturbances
Let the model of the system used for the computation of the optimal trajectories
be given by:
xm = f ( xm , um , t ;θ 0 )
xm ∈ X ⊂
where
fm :
n
×
m
×
→
n
is
n
the
state
vector,
(6.1)
um ∈ U ⊂
m
is
control
vector,
is the vector field representing the modeled dynamics, and θ 0 is
the vector of constant system parameters. Note that the subscript m is added to emphasize
the fact that these dynamics represent the modeled dynamics. The actual system
dynamics can be represented as:
x = f ( x, u, t ;θ ) + w(t )
where x(t ) is the actual state vector, u (t ) is the actual control vector, w(t ) :
(6.2)
→
n
is
the unknown but bounded disturbance vector, and θ is the vector of actual constant
156
system parameters. The error between the actual and the modeled state vector can be
written as:
t
t
0
0
x(t ) − xm (t ) = ∫ [ f ( x, u,τ ;θ ) − f ( xm , um ,τ ;θ 0 ) ] dτ + ∫ w(τ ) dτ
(6.3)
Defining δ x(t ) = x(t ) − xm (t ) and assuming that the function f (⋅) is Lipschitz continuous
[30] with respect to all of its arguments, the above relation can be used to determine an
upper bound on the state error as follows:
t
t
0
0
δ x(t ) ≤ ∫ ⎡⎣ L fx δ x(t ) + L fu u (t ) − um (t ) + L f θ θ − θ 0 ⎤⎦ dτ + ∫ w(τ ) dτ
(6.4)
where i represent the appropriate norm, such as the infinity-norm, and L fx , L fu , and
L f θ are the Lipschitz constants of the function f (⋅) for the arguments x(t ) , u (t ) , and θ ,
respectively, over their appropriate domains.
The adaptive trajectory following control law, developed in Chapter 3, can be
written as:
δ u (t ) = u (t ) − um (t ) = h ( x(t ) − xm (t ) )
where h :
n
→
m
(6.5)
is the control law (Eq. (3.11)) that maps the reference trajectory
xm (t ) and current state x(t ) to the control inputs required to drive the error δ x(t )
asymptotically towards zero. Under the assumption that this map is Lipschitz continuous,
a bound on Eq. (6.5) can be written as:
δ u (t ) ≤ Lh δ x(t )
(6.6)
Using this relation, the inequality (6.4) can be written as:
t
δ x(t ) ≤ ( L fx + L fu Lh ) ∫ δ x(t ) dτ + L f θ Θt + Wt
0
157
(6.7)
where w(t )
∞
= W and δθ
∞
= Θ . Using the Gronwell-Bellman Inequality [30], the
above relation can be written as:
δ x(t ) ≤ ( L f θ Θ + W ) t exp ⎡⎣( L fx + L fu Lh ) t ⎤⎦
(6.8)
Using relations (6.6) and (6.8) the upper bound on the control variations around the
nominal values can be written as:
δ u (t ) ≤ Lh ( L f θ Θ + W ) t exp ⎡⎣( L fx + L fu Lh ) t ⎤⎦
(6.9)
Although the bound given by the above inequality is simply an upper bound and
tighter bounds may be possible to estimate, it still provides some useful insights into the
constraint-tightening approach. It clearly shows that the control bound needs to increase
with time where the effect due to unknown disturbances and parameters will be greatest.
Thus, an effective way of tightening the constraints is to taper the control boundaries for
increasing time, rather than just lowering the bounds for all times. These variable control
constraints can be easily incorporated into the Legendre Pseudospectral method discussed
in Chapter 5. Another aspect of the size of these bounds is their dependence on the
“smoothness” of the dynamics and control laws, as captured by various Lipschitz
constants. Smoother dynamics result in lower values of the Lipschitz constants and
consequently lower variation in the control signals.
6.2.2. Trajectory Following Algorithm Description
The algorithm for trajectory following using the nonlinear adaptive control laws
developed in Chapter 3 can be described as follows:
•
Solve the optimal control problem for the formation, e.g. OTRPC (5.12) that
generates C1-smooth trajectories, with the following modifications:
158
¾ Constraint-tightening according to the bounds of unknown disturbances and
parameter variations (e.g. using (6.9)).
•
The optimal trajectory, along with the control history is used as the reference
trajectory for the trajectory following control law developed in Chapter 3 with the
following modifications:
¾ The constraints are relaxed to their original values.
¾ The dipole equations are solved such that the solution remains near the
optimal control values computed in the open loop problem. Thus, instead of
the AMMOP (4.50), the following optimization problem is solved during the
computation of the dipole vector:
N
{
min ∑ Tim (tk )
μ ( tk )
i =1
Subject to:
T
W
f m (p, μ) = a des
N
∑mf
i =1
}
+ μTi (tk ) Wmi μ i (tk ) + [μ i (tk ) − μ o (tk ) ] Wdi [μ i (tk ) − μ o (tk ) ]
i mi
(6.10)
(p, μ) = 0
where μ o (t ) is the optimal control history that is computed offline. This cost
function ensures that the control values remain near the optimal control history
and the formation remains on the optimal trajectory.
It should be noted that, due to constraint tightening, the resulting trajectories are
somewhat sub-optimal. But constraint tightening itself results in robustness of the
algorithm in the face of uncertainty. The constraint-tightening approach has been
successfully used in the linear MPC (Model Predictive Control) to enhance robustness
[67], [68]. A detailed theoretical discussion of effective ways of constraint tightening, for
159
nonlinear and constrained systems, is an open problem and beyond the scope of this
thesis. The fact that the algorithm presented above is practical and indeed robust is
demonstrated by the simulation results presented in the following section.
The 6DOF nonlinear simulation developed for LEO control demonstration was modified
to test the algorithm described above. The offline optimal trajectory was generated, with
a simple constant maximum current of 100 A in both the x- and y-coils, for a simple twosatellite reconfiguration problem in 2D. The trajectory following algorithm results are
shown in Figure 6.1 and Figure 6.2. The maximum current bound was relaxed to 110 A.
In these results, the exact Biot-Savart model [17] for the computation of the forces and
torques was used to simulate the formation while the far-field model was used for the
generation of the offline optimal trajectories as well as in the trajectory following
controller. Moreover, process noise was simulated as a random walk signal in the xdirection only, while sensor noise was also injected as shown in Figure 6.2. A
computational delay of 0.2 seconds was also used in the simulation.
Due to the presence of a disturbance force along the x-direction, the whole
formation was shifted to the left as shown in the trajecotry-plot in Figure 6.2.
Nevertheless, the relative distance error between the leader and follower satellites
remains relatively small as shown in Figure 6.1, thus showing the robustness of the
algorithm to both disturbances and model uncertainty. Also, note that the terminal values
of the reaction wheel angular momenta are not zero, as shown in Figure 6.1, although
they were constrained to be zero for the optimal trajectory. This is expected since the
160
formation is subject to external disturbances with nonzero net forces and also due to the
presence of measurement noise.
x - pdx
y - pdy
0
0
-0.05
m
0.05
m
0.05
0
200
-0.05
400
-3
x 10xdot - pdxdot
5
0
-5
0
200
0
0
x 10
0
200
Amp
Amp
200
time sec
Amp
-0.1
0
400
400
0
0
0
200
400
-5
5
0
0
0
200
time sec
400
0
200
400
hrwz Nms (follower)
200
-200
200
hrwz Nms (leader)
Iy A (leader)
Amp
0
400
5
Ix A (leader)
0
200
alpha error (deg) (follower)
0.1
200
-200
400
200
-200
0
Iy A (follower)
100
200
pdxdd
-0.2
0
Ix A (follower)
0
400
0
-5
400
200
0
200
-3
m/s2
m/s
5
alpha error (deg) (leader)
0.2
400
-5
0
200
time sec
400
Figure 6.1: Trajectory following results for a two-satellite formation using exact BiotSavart force-torque computation for the simulated system model, control and error
signals.
161
-20
0
200
time sec
-20
0
-3
5
x 10
200
400
time sec
sx for sat 1 (follower)
N
0
200
time sec
0
-0.02
0
200
time sec
0
200
time sec
400
Measurement error ry1
0.02
0
-0.02
0
200
time sec
400
6
100
x 10
200
300
time sec
sy for sat 1 (follower)
4
400
y (m)
0
-3
5
x-dist. (f)
leader (thick) and follower trajs.
0
-5
400
x 10
0
-5
400
Measurement error rx1
0.02
m
m
y w.r.t. leader and pdy (r dotted)
20
0
-3
5
0
-5
400
-3
x 10 x-dist. (l)
m
0
5
N
m
x w.r.t. leader and pdx (r dotted)
20
2
0
0
-5
-2
0
100
200
time sec
300
400
-4
-2
0
2
x (m)
4
6
8
Figure 6.2: Trajectory following results for a two-satellite formation using exact BiotSavart force-torque computation for the simulated system model.
6.3. Receding Horizon Control Formulation
In receding horizon (RH) control, also known as Model Predictive Control (MPC), an
optimal control problem over a finite horizon t H is solved and the resulting optimal
control is applied over a control horizon tC
t H . This process is repeated after every tC
seconds. The reasons for using a finite horizon instead of solving the full trajectory
planning problem are two-fold. First, the environment itself might be dynamic or
162
uncertain and second, the computational complexity can be prohibitively high for the full
trajectory planning problem in real-time. Usually a cost-to-go function is used to generate
the cost beyond the finite horizon t H and, at the same time, this cost-to-go function also
generates the target or terminal point for the finite horizon optimal control problem. See
References [69]-[72] for further background.
The advantages of using the RH-formulation are many and varied since it tries to
solve a real-world problem rather than a theoretical one such as LQR (Linear Quadratic
Regulator). This formulation can explicitly take into account all the constraints present in
the system. Moreover, it can handle dynamic changes in the environment, and at the same
time, generate fault tolerant algorithms to accommodate component failures [73]. These
advantages come at the cost of the computational complexity of the algorithm, which
depends on both the finite horizon optimal control problem complexity and that of the
cost-to-go function.
The RH control formulation has been successfully used for the real-time path
planning problem for UAVs by formulating the finite horizon optimal control problem as
a Mixed Integer Linear Program (MILP) and using the Dijkstra’s Algorithm to generate a
cost-to-go function [68], [74]. For EMFF, which belongs to the class of nonlinear and
constrained systems, both the finite horizon optimal control problem and the cost-to-go
function are computationally complex. In the following section, an RH formulation,
based on a simple cost-to-go function using the offline computed optimal trajectory, is
presented.
163
6.3.1. RH Formulation with Optimal Cost-to-Go Function
In this formulation the full optimal control problem for the electromagnetic formation,
OTRPC (5.12), is solved offline and the resulting C1-smooth trajectory and control are
used in the “optimal cost-to-go” function. Using this function, the RH-formulation is
shown in the form of a block-diagram in Figure 6.3.
Optimal Cost-to-Go
External
Disturbances
Optimal
Trajectory
x0, u0
RH
Controller
d
u
x
EMFF
Dynamics
Δ
+
x = Full state
u = Control Inputs
x0, u0 = Open loop optimal trajectory
d = Unknown external perturbations
Δ = Un-modeled dynamics
Sensor/ Estimation
Error
Figure 6.3: Receding Horizon formulation using optimal cost-to-go function.
The RH-controller shown in Figure 6.3 takes the terminal conditions from the
optimal cost-to-go function and solves the following optimal control problem online:
164
⎧
⎪
⎪
⎪
⎪
⎪
⎪⎪
OPRH ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩
min J = {x(t f ) − x 0 (t0 + t H )} H {x(t f ) − x 0 (t0 + t H )} + ∫ 1 + δ μ - μ 0 dτ
tf
T
t0
μ ,t f
Subject to:
Dynamics:
Initial Conditions:
x = f ( x, μ )
x0 = x(t0 ) = [x10
x 20
−μi max < μi (t ) < μ i max
x N 0 ]T
i = 1...N , t ∈ [t0 , t f ]
(6.11)
Collision Avoidance: min dij (t ) > Dmin ∀i ≠ j , i, j = 1...N , t ∈ [t0 , t f ]
RW Saturation:
C.M. Constraint:
h iRW (t ) < H i max
N
N
i =1
i =1
i = 1...N , t ∈ [t0 , t f ]
∑ mi xi 0 = ∑ mi xif
where t0 is the current time, H > 0 is a symmetric weighting matrix for the terminal
conditions, δ > 0 is a weighting constant, x0 (t ) is the optimal trajectory, μ 0 (t ) is the
optimal control vector that is computed offline and the rest of the symbols are as defined
in Chapter 5 for OTRPC (5.12). The purpose of adding the δ μ − μ 0 term in the cost
integral is to keep the control values near the optimal control. Note that the Optimal
Problem for Receding Horizon (OPRH) is formulated such that it can be solved in realtime using the Legendre Pseudospectral method discussed in Chapter 5. The real-time
solution is achieved by the following considerations:
•
A short value for t H ∼ 10sec is used which results in a smoother problem since,
during this short interval, the trajectory does not change much. Consequently a
smoother problem results in quicker convergence of the underlying PS
(Pseudospectral) approximation (see Theorem 1.1 in [75]).
•
The terminal conditions are included in the cost integral as a “soft constraint.”
165
•
Control is computed over a coarse grid ( N ≤ 10 ). This is possible since over a
short horizon a smoother trajectory can be approximated by a lower order
polynomial.
•
Using relaxed error tolerances in the solution of the NLP.
In order to test the algorithm a simulation was constructed in Matlab to implement the
RH-formulation for EMFF. The formation from Section 6.3.2 was used to test the
performance of the algorithm. The exact Biot-Savart law was used to compute the force
and torque in the nonlinear model while the far-field model was used to generate the
offline optimal trajectory and in the RH controller. A computation delay of 0.2 sec was
assumed in the simulation. A process noise was added in the simulation as a random walk
signal in the x-direction only, while sensor noise was added to both the x- and y-position
measurements.
In the finite horizon control formulation (6.11) simulation, the reaction wheel
saturation limit was increased from a value of 3 Nms (that was used in the offline
solution of the optimal control problem) to 3.5 Nms to account for external disturbances.
Also the control bounds were relaxed to 110A, from a value of 100A used in the
generation of offline optimal trajectories. The results are shown in Figure 6.4.
These results highlight the robustness of the algorithm to modeling errors and
external disturbances. Figure 6.5 shows the actual computation times required to run the
simulation on a 3.0GHz desktop computer. Although the simulation was built in Matlab
using the fmincon function, during most of the time the computation time is within 1sec
166
showing that using better coding and better NLP solvers (such as SNOPT) the algorithm
can be executed in real-time.
200
x error
-10
400
0
0.1
0
0
0
200
400
ix1 and ix10 (dotted)
Amp
0
-200
0
200
400
iy1 and iy10 (dotted)
-200
Amp
0
200
time sec
400
-1
0
200
400
HRW sat 1 (blue), sat 2 (g dash)
5
400
0
-0.1
-5
0
6
-200
4
0
200
400
iy2 and iy20 (dotted)
200
200
0
200
y error
0
200
400
ix2 and ix20 (dotted)
200
200
Amp
0
0.2
-0.2
200
leader
(sat
1)
2
0
0
200
time sec
400
-2
0
2 4
x (m)
2.5
2
1.5
1
0.5
0
0
50
100
400
-2
0
-200
0
Figure 6.4: RH-formulation simulation results.
Computation Time sec
Amp
0
Nms
0
alpha0 and alpha1 (red)
1
y (m)
m
-20
m
0
y w.r.t. leader and pdy (r)
10
m
m
x w.r.t. leader and pdx (r)
20
150
200
time sec
250
300
350
Figure 6.5: RH-formulation computation time during simulation.
167
6
It should be noted that the robustness to external disturbances is accomplished by
the constraint-tightening approach, also used in the adaptive trajectory following
algorithm. Comparing the RH-formulation results with those of the adaptive trajectory
following algorithm, it can be seen that the position errors are higher for the RHformulation. Nevertheless both algorithms exhibit robustness to external disturbances and
unmodelled dynamics.
6.3.3. Limitations and Possible Extensions
Using the optimal cost-to-go function in the RH-formulation has the limitation that it
requires offline computation of the computationally complex optimal control problem. In
addition, this algorithm is not suitable for a dynamic environment; although this is not a
limitation for EMFF since the space environment is well defined. Nevertheless, this
algorithm cannot be extended to include fault tolerance. A suitable cost-to-go function for
the electromagnetic formation can be based on the APFM (Artificial Potential Function
Method) described in Chapter 4. The Lyapunov Function, given by Eq. (4.19), can be
used as the cost-to-go function. In Chapter 4 it was shown that this function always
reconfigures the electromagnetic formation asymptotically, hence the cost-to-go function
using this method remains feasible during the maneuver. Note that APFM generates not
only feasible trajectories, it also computes the desired control signals to execute these
trajectories. Hence, the finite horizon optimal problem essentially optimizes this feasible
solution. The limitation of this approach is that the complexity of the APFM increases
with increasing number of satellites in the formation, as shown in Figure 4.13.
168
6.4. Comparison of APFM and Optimal Trajectory Following Method
In this thesis two methods of motion planning have been presented. The Artificial
Potential Function Method (APFM) generates collision-free trajectories for EM
formations as shown in Chapter 4. Chapters 5 and 6 present algorithms for generating and
following optimal trajectories for EMFF. It was mentioned earlier that APFM results in
suboptimal trajectories, although using Potential Function (PF) shaping (discussed in
Section 4.4) results in much improved results. The purpose of this section is to compare
the two methods in both the performance and computational complexity of the
algorithms.
In order to compare the two algorithms, rest-to-rest maneuver trajectories were
generated for random EM formations in 2D using both of the algorithms. The initial and
terminal conditions of these random formations were generated such that the distance
between any two satellites in the formation was greater than 3 m. The APFM trajectories
were used as the starting guess for the OTFM (Optimal Trajectory Following Method).
The optimal trajectories were generated using the Legendre Pseudospectral Method with
a grid size of N = 13. The Matlab constrained optimization function fmincon was used
with a tolerance of 1e-4 for the variables and 1e-2 for the constraints. These low values of
the tolerances were used so that the optimization would terminate in a reasonable time.
These results are shown in Figure 6.6.
169
Avegrage (over 5) rest-to-rest Maneuver times
80
APFM w/o PF shaping
APFM with PF shaping
Optimal Times
minutes
60
40
20
0
2
3
4
5
No. of satellites
Avegrage (over 5) rest-to-rest Maneuver Computation times
minutes
60
APFM w/o PF shaping
APFM with PF shaping
Optimal Times
40
20
0
2
3
4
No. of satellites
5
Figure 6.6: Comparison of APFM and OTFM trajectories.
Figure 6.6 shows clearly that using PF shaping results in considerable
improvements in the performance of the APFM algorithm. As compared to the optimal
trajectories, the APFM trajectories take approximately 2 to 4 times more time to complete
the maneuver. Although in the above results the parameters of the PF shaping algorithm
can be further fine-tuned to decrease the maneuver times (since these parameters need to
be tuned for each formation and it takes considerable time so rather conservative values
of the parameters were used in the above results). Nevertheless, these results highlight the
important fact that the PF shaping algorithm is a promising technique that can be used to
generate feasible and practical trajectories for EM formations.
170
It can be seen from Figure 6.6 that the APFM runs in real time to generate the
collision-free trajectories for the EM formations and the complexity of the algorithm
increases as O(n) where n is the number of satellites in the formation. Although in the
OTFM the open loop optimal trajectories are computed offline, the complexity of the
OTFM open loop trajectory algorithm also increases as O(n). In the OTFM, as discussed
earlier, a finite horizon optimal control problem is solved at each time step whose
computational complexity is not shown in Figure 6.6. From a practical point of view,
implementing OTFM in real-time requires much more computational power as compared
to the APFM, since APFM generates control inputs based on a closed-form Lyapunov
function and the solution of the dipole equations. It is the latter computation (i.e., the
dipole equation solution) that increases the computational complexity of the APFM
algorithm. Nevertheless, the complexity of the dipole equation solution algorithm is
much lower as compared to the online optimization problem of the OTFM.
Another important aspect for EM formation reconfiguration is the angular
momentum management during the maneuver. APFM does not allow any direct control
over the reaction wheel angular momentum during the maneuver (although simulations
can be used to ensure that the RWs do not saturate). On the other hand, OTFM allows
complete control over the RW saturation, and the maximum angular momentum that a
RW can store can be specified in the computation of the trajectories.
From the above discussion it is clear that OTFM is the most general method for
trajectory generation for nonlinear and constrained systems, such as EMFF, but it is much
more computationally expensive to implement as compared to the APFM. This
comparison is summarized in Table 6-1.
171
Table 6-1: Comparison summary for OTFM and APFM
Comparison Measure
OTFM
APFM
Computational Complexity
Very High
Low
Maneuver Time Optimality
Near Optimal
Sub-optimal (~ factor of 3)
Reaction Wheel Saturation
Full control
No control
6.5. Summary
In this chapter two algorithms for following optimal trajectories were presented. One
algorithm is based on the nonlinear adaptive trajectory following algorithm presented in
Chapter 3 for LEO EM formation control. This algorithm models the unknown external
disturbances and model uncertainties as slowly varying parameters and has very low
computational burden. The second algorithm is based on the receding horizon control
formulation. This formulation solves a finite horizon optimal control problem at each
control time step and applies the computed control values until the next control step, at
which point the solution of the finite horizon optimal control problem is repeated. The
robustness in the RH-formulation is accomplished by tightening the constraints in the
solution of the offline optimal control problem. Thus this chapter, along with Chapter 5,
presents a complete method, called OTFM (Optimal Trajectory Following Method), of
implementing optimal trajectories for nonlinear and constrained systems with specific
application to EMFF. This method is also compared to the motion planning algorithm
using APFM (Artificial Potential Function Method) that was presented in Chapter 4. This
comparison reveals that the APFM, with Potential Function Shaping (discussed in
172
Section 4.4) is a promising technique that can be used to generate collision-free
trajectories, which are sub-optimal by a factor of around 2, for EM formations at a low
computational burden as compared to OTFM.
173
Chapter 7
Conclusions
7.1. Summary and Contributions
Electromagnetic formation Flying (EMFF) is a novel concept that uses high temperature
superconductors to create magnetic dipoles in order to generate forces and torques
between the satellites in a formation. Using EMFF, all the relative translational degrees
of freedom can be controlled without expenditure of any consumables and consequently
the mission lifetime becomes independent of the fuel available on board. This comes at
the cost of increased complexity in the dynamics and control problem associated with the
formation flying. The key objectives of this thesis were to show that the EMFF is a
feasible concept from the dynamics and control point of view, and to develop practical
algorithms that can be implemented in real-time for the control of general
electromagnetic formations.
In the following paragraphs, the chapter-wise summary and key contributions of
this thesis are presented:
•
Chapter 1 introduces the concept of EMFF and shows that EMFF has distinct
advantages over conventional thruster based propulsion for certain formations in
LEO.
175
•
Chapter 2 derives the dynamic models necessary to simulate and control EM
formations in both LEO and deep-space. Lagrangian formulation is used in the
derivation of the nonlinear translational models that includes the J2 perturbations
as well and allows easy additions of higher order terms. The key contribution of
this chapter is that it presents 6-DOF models suitable for simulation and control
design of general EM formation both in LEO and deep-space.
•
Chapter 3 presents a general control framework in which control laws for specific
missions can be designed. Using this framework, 6-DOF adaptive trajectory
following control laws for general EM formations have been derived. These
control laws account for the model uncertainty due to the far-field approximation
and the uncertainty in the Earth’s B-field by using slowly varying parameters. The
key contribution of this chapter is that it presents a dipole polarity switching
control law for the management of angular momentum in LEO for general EM
formations. Also using 6DOF nonlinear simulations, the feasibility of these
control laws is shown for LEO formations.
•
Chapter 4 presents the Artificial Potential Function Method (APFM) with specific
application to EMFF as a complete algorithm for motion planning of general EM
Formations. The key contribution of Chapter 4 is that APFM and Lyapunov
theory are used to generate a trajectory planning algorithm that can be used to
generate collision-free trajectories for general EM formations. This algorithm is
computationally efficient and can be used to generate trajectories for moderately
sized (N~10) EM formations in real time. This chapter also presents a new
176
method of solving the dipole equations, namely dynamic inversion through
optimization for EMFF.
•
Chapter 5 presents optimal trajectory generation algorithms for general EM
formations using two direct solution algorithms, namely Direct Shooting and the
Legendre Pseudospectral (LPS) method. The key contribution of Chapter 5 is that
it gives an efficient formulation of the optimal control problem using the LPS
method for EMFF by exploiting the Newtonian dynamics formulation instead of a
state-space representation.
•
Chapter 6 presents two methods for following optimal trajectories for EMFF. The
first method uses the adaptive control laws developed in Chapter 3. The second
method is based on the Receding Horizon (RH) formulation. The key contribution
of this chapter is that it formulates the RH problem for EMFF such that it can be
solved in real time. Thus Chapter 5 and 6 present practical algorithms for
implementing optimal trajectories in real-time for nonlinear and constrained
systems. Moreover Chapter 6 also compares the APFM and Optimal Trajectory
Following Method (OTFM) for EMFF and the pros and cons of both methods are
highlighted with application to nonlinear and constrained systems.
7.2. Future Directions
For the generation and implementation of feasible trajectories for the class of nonlinear
and constrained systems, this thesis presents two methods, namely APFM (Chapter 4)
and the Optimal Trajectory Following Method (OTFM) (Chapter 5 & 6) with their
comparison given in Chapter 6. One possible area of research is to reduce the
177
convergence time of the APFM algorithm so that the resulting trajectories are nearer to
the optimal trajectories. Some of the ideas to achieve this objective are presented in
Chapter 4 (i.e., the Potential Function Shaping method). Second, in the RH-formulation
the cost-to-go function was based on optimal trajectories; and was computed offline.
Using this “optimal cost-to-go function,” results in a RH-formulation that is less
appealing, especially from real-time fault tolerance point-of-view. A possible better
formulation can be to use the APFM Lyapunov function as the cost-to-go function in the
RH-formulation. This framework could give rise to anytime motion planning [78] for
general EM Formations. In the anytime motion planning algorithm, a feasible trajectory,
with some minimal desired properties, is generated quickly and made available to the
system; while in the remaining time this trajectory is optimized to improve performance.
To have real world demonstration of the workings of the control laws developed
in this thesis, these laws need to be tested on the EMFF testbed on a flat floor. Both the
optimal trajectories and the APFM can be tested on the EMFF testbed by integrating the
existing code with the SNOPT software for embedding the controller on the testbed DSP.
178
Appendix A:
Matlab Tools for Simulation of Electromagnetic
Formations
A nonlinear simulation for a general N-satellite electromagnetic formation has been
prepared in the Matlab and Simulink environment (see Figure A.1). This simulation is
used to test the nonlinear control laws for the formation. The different functions and
blocks used in the simulation are described in the following sections.
Nveh3dFT:
A Matlab mex function, written in C-language that generates the force and torque
on each Electromagnetic Satellite in a general N-satellite formation. This function takes
the current state (position and orientation) and control inputs (current in each coil) of
each satellite in the formation. This function implements both the far-field model (much
faster) and the exact model using the Biot and Savart Law. The user has the option to
select either of the models in force and torque computations. This function assumes that
each satellite has three orthogonal circular coils and different coil parameters such as area
and number of turns are supplied by the user. This function can be used in any general
dynamic simulation of general electromagnetic formations.
3D Dynamics S-Function:
This Simulink S-Function implements the general nonlinear dynamics for an
electromagnetic formation in deep-space as well as for Low Earth Orbit (LEO)
179
operations. The dynamics include the translational as well as attitude dynamics. The deep
space translational dynamics are simply double integrator dynamics with Electromagnetic
actuation while the LEO dynamics are in the ECI (Earth Centered Inertial) reference
frame and also include the J2-perturbations. The attitude dynamics are based on Euler
equations and use quaternions and include the gyro-stiffening effect as well. In these
dynamics each satellite is assumed to have three orthogonal reaction wheels aligned with
the principle axes of the satellite for simplicity. This function takes the control inputs and
the process noise perturbations as inputs and generates the state vector of the whole
formation. This function also accounts for the disturbance torque on each satellite due to
the Earth’s magnetic field and currently a simple dipole based Earth’s magnetic field
model is being used for simplicity (which can be enhanced readily using the available
WMM or IGRF codes).
Controller S-Function:
In order to test the viability of using the electromagnetic formation concept in
LEO, a nonlinear adaptive control scheme has been developed and implemented in the
form of a Simulink S-function to control an N-satellite electromagnetic formation in
LEO. This controller implements both the translational and attitude control for a general
N-satellite formation in a circular orbit around Earth. This function takes the current
formation state and desired trajectory as inputs and generates the coil currents and
reaction wheel torques as outputs.
180
pf x
xt
pf y
To Workspace6
pf z
inputs
xt
ptx
pty
3D NL Model
ptz
Process Noise
Model
u
x
To Workspace5
y
z
q1
Saturation
Ix
q2
q3
Saturation1
Iy
q4
xd
Saturation2
yd
zd
Iz
Tx
contLEO
Ty
em
w1
w2
w3
omega1
omega2
sensors
Sensors Model
omega3
the1
time
Clock
the2
Tz
To Workspace
the3
pdx
pdy
pdz
pdxd
ahat
pdy d
To Workspace2
current time
pdzd
pdxdd
pdy dd
st
pdzdd
To Workspace1
S-Function
Desired
Trajectory Generator
Figure A.1: Nonlinear 3D Simulink simulation for electromagnetic formation.
181
Appendix B: Precision Formation Control using EMFF
A relevant question about formation flying is the ability of the control system to maintain
the formation within a narrow tolerance during an observational mode of the system.
Needless to say, this question is more pertinent for the nascent and innovative concepts
like EMFF. In this section we will show that the EMFF concept does not have any
inherent limitation, from a theoretical point of view, and can meet the precision FF
requirements. This will be shown by construction for the Gen-X mission (see Reference
[42] for an introduction to the Gen-X mission).
B.1 Gen-X Precision FF Requirements
For the Gen-X mission the pointing accuracy drives the precision formation flying
tolerances. The position control tolerances between the optics and detector are assumed
to be 0.1 to 0.3 mm in separation and several mm in the plane perpendicular to the optical
axis. It is pointed out that this separation can be maintained by using staged control if the
formation is not capable of providing the necessary precision. Overall formation pointing
accuracy requirements can be stated as:
•
•
•
•
•
Boresight pointing accuracy
Boresight pointing knowledge
Drift or jitter
Detector position Error along boresight
Detector position Error normal to boresight
183
< 1 arc-second
< 0.5 arc-second
< 1 arc-second/minute
~ 0.1 mm
~ 1 mm
B.2 Problem Formulation
It is assumed that the primary module is pointed to the desired orientation, satisfying the
boresight pointing requirements, using the primary reaction wheels and other necessary
instruments. The task of the formation control subsystem is to keep the detector module
within the specified tolerances. The detector formation flying for the Gen-X mission will
be studied for the following scenario:
1) The spacecrafts are not slewing and are in the observation mode.
2) The control system will be formed as a regulator to maintain the formation within
the given error bounds.
3) Only 3-DOF (Degrees of Freedom) control is considered in this study for
simplicity. This essentially assumes that the motion of the system is restricted to
the slewing plane and system is infinitely stiff to disturbances normal to the
slewing plane. This can be easily extended to full 6-DOF control setup by
designing the controller for each axis independently as the linearized dynamics
are decoupled for EMFF in the case of 3-orthogonal superconducting coils on
either of the modules.
4) For the sake of this analysis the primary is assumed to have one coil and it is
maintained at a specified constant current for simplification of the control law.
The primary module also has two orthogonal reaction wheels to maintain its
attitude.
5) The detector module has two orthogonal coils and two orthogonal reaction wheels
for control.
6) System model/parameters are assumed to be known accurately.
184
7) The measurement bandwidth is much higher than the required closed loop
bandwidth of the system. This essentially means that we can approximate the
measurement instruments as having a unity transfer function in the range of the
frequencies of interest.
The origin of the global coordinate system is assumed to be at the c.m. of the system. The
detector module is placed at a focal length distance from the primary module along the xaxis as shown in Figure B.1. As described earlier, we have separated the formation flying
problem from the telescope pointing problem and we will only consider the former
problem, i.e., the problem of keeping the detector module within tolerances for position
and attitude given the location and orientation of the primary. Thus, in this formulation
the formation may drift under external force disturbances but the relative orientation and
distance between the primary and detector modules is to be kept within bounds so that the
telescope keeps on pointing to the target within the boresight pointing accuracy and jitter
tolerance.
y
x
detector
primary
Figure B.1: Coordinate setup for control.
185
B.3 Regulator Setup
The regulator setup for the problem is shown in Figure B.2. In this setup, as described
earlier, it is assumed that the sensors have enough bandwidth such that the sensor
dynamics can be neglected. The plant P is assumed to be linearized at the nominal
operating point, F is the controller, e is the vector of errors, d are the disturbances
referred to the actuator input, y is the vector of outputs and v is the sensor noise vector.
Weighting W1 and W2 are shaping functions that are used to weight the error as a function
of frequency, i.e., we use these in such a way as to minimize the error in the desired range
of frequencies rather than the whole spectrum which would be wasteful. Costs z1 and z2
are used in the optimization setup to select the appropriate controller that minimizes these
costs. This setup can be represented in a canonical setup for state-space design as shown
in Figure B.3.
+
_
d
0
+
F
+
+
W1
e
y
P
_
W2
z2
+
v
+
Figure B.2: Block diagram for the regulator setup.
186
z1
z
w
P
u
y
F
Figure B.3: Standard State-Space setup for the regulator problem.
The plant can be described in the standard form by the following set of equations:
x = Ax + B1w + B 2u
z = C1 x + D11 w + D12u
(B.1)
y = C 2 x + D 21 w + D 22u
In this problem setup we want to minimize some form of norm from the noise vector w to
the cost vector z, with the constraint that the closed loop system has stable dynamics and
acceptable transient characteristics. A well known solution to this problem is the H∞
controller design based on the minimization of the H∞ norm from w to z, i.e.,
Twz
∞
<γ
(B.2)
where γ is a positive number that we try to minimize. To see that this is indeed a good
way to design the controller for this problem, we can write the error signal as:
e = − SPd − ( I − S ) v
where S is the sensitivity function defined as:
187
(B.3)
S = ( I + PF ) −1
(B.4)
Clearly to minimize the error due to disturbances, the sensitivity matrix should be “small”
in some sense. Thus, the sensitivity matrix would be small if the smallest singular value
of the loop gain matrix PF is much greater than unity or:
σ ( PF ) >> 1
(B.5)
Since the noise vector affects the error via the complementary sensitivity matrix making
S “small” would result in direct injection of the sensor noise to the error signal. This is a
fundamental tradeoff of the feedback control. Therefore to have good disturbance
rejection one must have a very low sensor noise in the frequency range of interest. At
higher frequencies the sensitivity matrix is almost unity thus the high frequency sensor
noise would not have a significant effect on the error dynamics.
From Figure B.2, we can write Eq. (B.2) in terms of the sensitivity matrix as follows:
W1 ( I − S )⎤
⎡ − W1 S
⎢− W ( I − S ) − W SP ⎥
2
2
⎣
⎦
<γ
(B.6)
∞
and minimizing the H∞ norm of the matrix Twz would make sure that each of the
submatrix transfer functions in Twz is bounded from above by Twz
∞
, therefore the
weighted sensitivity matrix is minimized as desired. Weighting matrices W1 and W2 are
selected in an iterative fashion until the requirements on the sensitivity matrix are
satisfied.
188
B.4 Plant Dynamics
For the Gen-x formation flying using EMFF the relative dynamics of the detector module
in the slew plane with respect to the coordinates fixed at the system c.m. can be described
by the following equations:
⎡ Fdpx ⎤
⎥
⎢
M eq ⎥
⎢
x
⎡ d⎤
⎢ y ⎥ = ⎢ Fdpy ⎥ + disturbances
⎥
⎢ d⎥ ⎢ M
eq
⎥
⎢⎣α d ⎥⎦ ⎢
⎢ Tdp − Tdrw ⎥
⎥
⎢
⎦
⎣ Id
(B.7)
where:
xd = Position of the detector along x-axis,
yd = Position of the detector along y-axis,
αd = Angle of the detector module body fixed x-axis w.r.t. global x-axis,
Fdpx = Force on the detector due to primary along x-axis,
Fdpy = Force on the detector due to primary along y-axis,
Tdp = Torque on the detector due to primary about z-axis,
Tdrw = Torque on the detector due to the detector reaction wheel,
Meq = Equivalent mass of the primary-detector system =
M DM P
,
MD + MP
Id = Moment of inertia of detector about z-axis.
Using the far field approximations for the magnetic forces and torque, the
linearized equations of motion can be written as:
189
⎡ xd ⎤ ⎡ 0
⎢ x ⎥ ⎢0
⎢ d⎥ ⎢
d ⎢ yd ⎥ ⎢ 0
⎢ ⎥=
dt ⎢ yd ⎥ ⎢⎢ 0
⎢α d ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣⎢α d ⎦⎥ ⎣ 0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⎡
⎢ −3 μ μ
0 px0
⎢
0 ⎤ ⎡ xd ⎤
⎢ 2π M eq d 04
⎢
⎥
⎥
x
0
⎢
⎥⎢ d ⎥ ⎢
0
0 ⎥ ⎢ yd ⎥
⎢
⎥⎢y ⎥+ ⎢
0
0
d
⎥⎢ ⎥ ⎢
1 ⎥ ⎢α d ⎥ ⎢
0
⎥⎢ ⎥
0 ⎦ ⎣α d ⎦ ⎢
⎢
0
⎢
⎣
0⎤
0
0
0
3μ 0 μ px0
4π M eq d
4
0
0
−
μ 0 μ px0
2π I d d 03
⎡ 0
⎥
⎢ 1
⎢
0⎥
⎥
⎢ M eq
⎥
⎢
0 ⎥ ⎡ Δμ dx ⎤ ⎢ 0
⎥ ⎢ Δμ ⎥ + ⎢
dy
⎥ ⎢ 0
0 ⎥⎢
⎢
T
⎥ ⎣ drw ⎥⎦ ⎢
⎥
⎢ 0
0⎥
⎢
⎢
1⎥
⎥
⎢ 0
Id ⎦
⎣
0
0⎤
0
0⎥
0
1
M eq
0
0
⎥
⎥
⎥
0 ⎥ ⎡ dx ⎤
⎥ ⎢d ⎥
0 ⎥⎢ y⎥
⎥ ⎢⎣ dα ⎥⎦
0⎥
⎥
1⎥
I d ⎥⎦
where:
μ0 = Permeability of free space = 4π·10-7 N/A2
μpx0 = Constant magnetic moment of the primary coil
Δμdx = Magnetic moment of x-coil on the detector module
Δμdy = Magnetic moment of y-coil on the detector module
d0 = Nominal separation between the primary and detector
dx, dy and dα = Actuator referenced disturbances acting on the detector.
From these linearized dynamics we can see that the telescope has decoupled
control except for angular motion. Since the bandwidth of angular control is much higher
as compared to that of the EMFF coils, for the purpose of showing the feasibility of using
EMFF for precision formation flying we can treat the dynamics as decoupled. This would
result in much simplified analysis since we only have to show that a controller can be
designed that would keep the tracking error within bounds for each axis separately.
Considering the dynamics for x-axis we obtain:
0
⎡
⎤
⎡ 0 ⎤
d ⎡ x d ⎤ ⎡ 0 1 ⎤ ⎡ x d ⎤ ⎢ − 3μ μ ⎥
0
px0
+
=
Δμ dx + ⎢ 1 ⎥ d x
⎢M ⎥
dt ⎢⎣ x d ⎥⎦ ⎢⎣0 0⎥⎦ ⎢⎣ x d ⎥⎦ ⎢ 2πM d 4 ⎥
eq 0 ⎦
⎣ eq ⎦
⎣
190
(B.9)
(B.8)
B.5 Disturbance Sources
The main contributors to the disturbances acting on the system are as follows:
•
•
•
Solar pressure
Quantization noise
Thermal noise
Solar pressure on the telescope results in external torques and forces acting on the
system. At L2 the value of radiation pressure is approximately 4.5x10-6 N/m2. Thus an
external disturbance force of the order of few micro-Newton would act on the system.
This force would be almost constant during an observation and thus act like a bias
disturbance.
Quantization noise would result from the finite discrete levels of the DAC (Digital
to Analog Converter) present in the control system. This noise depends on the number of
bits used to represent the analog value in the digital to analog conversion system.
Thermal noise is the noise that would be present at the output of the coil driver
amplifier. Since the bandwidth of the coil is low and a well designed amplifier would
have thermal noise well below quantization noise, its effect will be only secondary to that
of the quantization noise.
B.4 Controller Synthesis
The controller is synthesized for each axis separately as discussed in the previous section
on the dynamics. In this section we will present the synthesis of a single axis controller
for the x-axis using Matlab tools for H∞ synthesis. Using Figure B.2 as a guideline, the
problem for controller synthesis is setup using simplified one-axis Eq. (B.9) as shown in
Figure B.4. The weighting functions for error W1 and for control W2 are shown in Figure
191
B.5. Error in position is weighted so that the controller will try to minimize the error
within a band of 1 Hz while for higher frequencies the weighting itself makes the error
low so the controller does not waste its effort in minimizing error at higher frequencies.
Similarly control effort weight is selected so that using control at lower frequencies is
less costly as compared to higher frequencies. In both cases Bessel filters of order 2 were
used to synthesize the low pass filter and a high pass filter for W1 and W2 respectively.
The cut-off frequency for error weighting is selected such that the error remains below
the required threshold within the desired bandwidth. Similarly the cut-off frequency for
the control weighting is selected to be below the bandwidth of the coil-driver circuit.
An integrator is placed in series with the plant so as to make the steady state error
zero for bias disturbances. This integrator is later placed in series with the synthesized
controller and becomes a part of the controller. Using this setup an H∞ controller was
synthesized with the resulting closed-loop transfer functions shown in Figure B.6.
Figure B.4: H∞ controller design setup.
192
Bode Magnitude Diagram
20
W1
W2
0
Magnitude (dB)
-20
-40
-60
-80
-100
-120 -2
10
-1
10
0
10
w (rad/sec)
1
10
2
10
Figure B.5: Weighting filters for error W1 and control effort W2.
We can see from Figure B.6 that the closed-loop transfer function from actuator
referenced disturbances, namely w2, to error z1 is well below -25 dB within the designed
bandwidth of 1 Hz. Therefore all the disturbances are attenuated considerably as also
shown by a time simulation (see Figure B.7) of the closed-loop system with the
synthesized controller. In this simulation a constant bias force of approximately 0.4 mN
was used. It was assumed that the current driver had large thermal and quantization noise
(with a variance of ~ 2 Amperes, uniformly distributed). As can be seen from Figure B.7,
even large quantization error resulting in considerable noise current still results in a very
small closed loop error well within the desired bounds.
193
Bode Magnitude Diagram
50
0
Magnitude (dB)
-50
-100
-150
-200
-250
Tw1z1
Tw1z2
Tw2z1
Tw2z2
-300
-350 -8
10
-6
10
-4
-2
10
10
Frequency (rad/sec)
0
10
2
10
Figure B.6: Closed-Loop TFs for H∞ synthesized controller.
B.7 Conclusion
From the results in the previous sections we conclude that a controller can be designed
for EMFF that would keep the formation within the desired bounds of error for a
reasonable bandwidth. Therefore the EMFF concept does not have any inherent
limitations as far as precision formation flying is concerned.
194
dist. force (mN)
disturbance force vs time
2
1
0
-1
50
40
100
150
200
Ix
Ic
IQ
A
20
0
x-Error (mm)
-20
50 -3
x 10
1
250
100
150
200
x-Position Error in mm vs time
250
0
-1
-2
50
100
150
Time (sec)
200
250
Figure B.7: Time response of the closed-loop system to disturbances. Ix is total actuator
current, Ic is commanded current and IQ is the current resulting from quantization
process.
195
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203

Dynamics and Control of Electromagnetic Satellite Formations

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